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CARPENTER'S NEW GUIDE: \ 

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coMPKETE BOOK dr kzhtes 

TREATING FPLLY OI* 

' SOFFITS, BRICK AND PLASTER GROINS, NICHES OF EVERY DESCRIPTION, 
SKY-LIGHTS, LINES FOR ROOFS AND DOMES ; • 

WITH A GREAT VARIETY OP 

Duignsfor Roofs, Trussed Girders, Floors, Domes, Bridges, iSfC.} Stair 'cases and Hand-rails of various 
Constructions } Jingle Bars for Shop Fronts, «^c. ; and^aking Mouldings ; 

WITH MANY OTHER THINGS ENTIRELY NEW: 

THE WHOLE FOUNDED ON TRUE GEOMETRICAL PRINCIPLES ; 

THE THEORY AND FRACTICE WELL EXPLAINED AND FULLY EXEMPLIFIED ON 

EIGHTY-FOUR COPPER-PLATES: 

INCLUDING ^^^« 

' SOME OBSERVATIONS AND CALCULATIONS ON THE STRENGTH OF TIMBER. 



Author of the Carpenter and Joiner's Assistant', The Student's Instructor to the Five Orders, &c. 
^19 ^^^' .-;. .-:; i* TESTTB EDITION. 

T'^rilwaED Bk iOHN GRIGO^ No, y, NORTBj FGURTDH STREET. 

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PREFACE. ^^^ 



To a book intended merely for the use of Practical Mechanics, much Preface 
is not necessary: — It is proper, however, to say, that whatever rules by previous 
authors have on examination proved to be true and well explained, these have been 
selected and adopted, with such alterations as a very close attention has warranted 
for the more easily comprehending them, for their greater accuracy or facility of ap- 
plication : added to these, are many examples which are entirely of my own inven- 
tion, and such as will, I am persuaded, conduce very much to the accuracy of the 
work and to the ease of the workman. 

The arrangement of the subjects in this work is gradual and regular, and such as 
a student should pursue who wishes to attain a thorough knowledge of his profes- 
sion : and as it is Geometry that lays down all the first principles of building, mea- 
sures lines, angles, and solids, and gives rules for describing the various kinds of 
figures used in buildings ; therefore, as a necessary introduction to the art treat^k 
of, I have first laid down, and explained in the terms of workmen, such problems or 
Geometry as are absolutely requisite to the well understanding and putting in prac- 
tice the necessary lines for Carpentry . These problems duly considered, and their 
results well understood, the learner may proceed to the theoretical part of the subject, 
in which Soffits claim particular attention ; for, by a thorough knowledge of these, 
the student will be enabled to lay down arches which shall stand exactly perpendi- 
cular over their plan, whatever form the plan may be : on this depends the well 
executing all groins, arches, niches, &c. constructed in circular walls, or which stand 
upon irregular bases; wherefore the importance of rightly understanding these I 
cannot too much insist upon, their construction being so various and intricate, and 
their uses so frequently required. The two ^plates of cuneoidal or winding soffits 
are new, and are constructed in a more simple and --m ore accurate manner : yet this 



V- 



iy , PREFACE. * 

method is only a nearer approximation to truth than the former one; the surface of 
a conchoid cannot he developed ; that is, it cannot he extended on a plane : it is 
therefore ahsurd to look for perfection on this suhject. 

The next suhject which regularly presents itself is Groins ; for the construction 
of which there will he found many methods entirely new ; and hesides the common 
figures, I have shown many which are difficult of execution, and not to be found in 
any other author. I have displayed a variety of methods for constructing spherical 
niches, a form more frequently wanted than the elliptic, which only has yet been 
explained. 

Among the various methods for finding the Lines for Roofs, I have given an en- 
tire new one for finding the down and side bevels of purlines, so that they shall 
exactly fit against the hip rafter; and by the same method the jack rafter will be 
made to fit. 

Of Domes and Polygons, I have shown an entirely new method for finding their 
covering, within the space of the board, thereby avoiding the tedious and incommo- 
dious method of finding the lines on the dome itself, as has been always practised 
heretofore ; also a method for finding the form of the boards near the bottom, when 
a dome is to be covered horizontally. Of dome-lights over stair-cases, or in the 
centre of groins, a rule upon true principles is given, for finding their proper curve 
gainst the wall, and the curve of the ribs ; this has never before been made public. 

Having gone thus far in the Art of Carpentry, it is necessary for me to say, by 
way of caution and guard to the ardent theorist, that there are some surfaces whieh 
cannot be developed ; such as spherical or superoidical domes, where their coverings 
cannot be found by any other means than by supposing the curved surface to become 
polygonal; in which case such domes may be covered upon true principles, as may 
be demonstrated. Let us suppose a polygonal dome inscribed in a spherical one; 
then, the greater the number of sides of the polygonal dome, the nearer it will coin- 
cide with its circumscribing spherical one. Again, let us suppose that this polygonal 
dome has an infinite number of sides; then, its surface will exactly coincide with 
the spherical dome, and therefore in any thing which we shall have occasion to 
practise, this method will be sufficiently near ; as for example, in a dome of one hun- 
dred sides, of a foot each, the rule for finding such a covering will give the practice 
so very near, that the variation from absolute truth could not be perceived. 



i 



J ^ 



PREFACE. 



Having gone through the constructive part of Carpentry, I next proceed to ex- 
amples showing the best forms of floors, partitions, trusses for roofs, truss girderf^ 
domes, &c. which shall resist their own weight, or the addition of any adventitious 
load. 

In that nice and elegant branch of the Building Art, called Joinery, Stairs and 
Hand-rails take the lead; and notwithstanding the great importance of this subject, 
I am sorry to find it has been treated, by all authors, without the governing princi- 
ple of science. For Stair-cases, in general, I have laid down correct methods, 
founded on the most obvious principles, and which, since the first publication of 
this work to the present time, I have the satisfaction to say, have been put in prac- 
tice, and found to answer well. 

In this Edition, a new and correct method for ascertaining the spring of the 
plank has been introduced instead of the former, which though very near was not 
exactly true, except when the plane of the plank was at right angles to the chord 
plane. I have likewise shown the construction of rails with butt joints, a method 
much stronger and more frequently practised than splice joints. This occasioned 
the introduction of plates 59, 65, 66, 67, 68, and the introduction of Jig. C m 
plate 70. 

The butt joints shown in plates 59 and 66, and the subjects exhibited in plates 56 
and 57, are not to be found in any former Edition. ' 

In Plate 56, a cylinder is represented equal to the thickness of the rail, showing 
both the concave and convex sides, with the development of the falling mould, 
and the orthographical representation of the steps, and the vertical surfaces of the 
wreath, coiling round the cylinder, in order to give a clear idea of forming the 
twist of the rail, in the execution of the work- 

The introduction of plate 57, in this Edition, was indispensable, in order to as- 
certain the exact thickness of stuff, as no method has ever yet been shown by which 
this requisite might be obtained. It would be rather troublesome to the workman 
to make an orthographical representation at large on a floor where, perhaps, he 
might not have room; but this might be effected by a scale on paper, and thereby 
make it an evening's amusement, and thus much trouble would be saved. 



n PREFACE. 

^ Plate 69 has been introduced instead of one in the former Edition. The de- 
Mopment of figures 2 and 3, exhibits the full effect both of the string and rail, 
anq[ enables the reader to comprehend the intentions of the author, and the advan- 
tagfe to be obtained in point of ease and regularity, in the form both of the stair 
andkhe hand-rail. 
1 

^'liis principle of gradating the steps, and forming a graceful curve on the soffit, 
will hk as useful to the Mason, in the execution of stone stairs, as to the Joiner. 
/\ ' ^^ 

In^yiiiis Edition, besides the six new plates, the whole have been carefully ex- 
amined, and some re-engraved, and every part of the explanation which did not 
appear sufficiently clear, has been better expressed, and the arrangement has been 
improved. 

To conclude ; as I pretend not to infallibility, I hope to be judged with candour, 
being always open to conviction, from a knowledge of the difficulty and intricacy of 
science ; yet I hope that my labours -may be of some use to others in shortening the 
road, and smoothing the path through which, for many years, I have been a perse- 
vering traveller for knowledge : I shall then be satisfied, and not deem my time 
mispent if my labours tend to the public good. 

P.NICHOLSON. 



A\ 



CONTENTS. 



PRACTICAL GEOMETRY. 

Page 
Definitions .-.----- ---10 

Problems _-...----- 13 

CARPENTRY. 



.4 



Linings of Soffits - - - - - - - - - - 29 

Kirb Lights for Church Work - - - - - - - - 34 

Groins - - -- - - -- - --S6 

Niches ----------- 47" 

Angle Brackets for Plaster Cornices - - - - - - -50 

Pendentives ---------- 51 

Roofing, Plain and Spherical - - - - - - - -54 

Strength of Timber .-......- 63 

Designs for Roofs - - - - - - - - - -79 

JOINERY. 

Hand-railing and Stair-casing - - -- - - - -'85 

Diminishing Columns - - - - - - - - -110 

Cylindro-cylindric Sash Work, or Circular-headed Sash Work, in Circular Walls - 111 
Angle Bars for Shop Fronts --------- 113 

Raking Mouldings -.---_..- ii4 

The method of proportioning Cornices - ■= - . . - . - ib. 

Mouldings upon the Spring - - - -.- - - - 115 

Sky-lights - - - - : '. - - - - - - 116 

CONCLUSION. 

The Ellipsis - - - - , - - . . . - 119 

Raking Mouldings - -.-v^- ,. . . _ 120 

DiminisluDg Columns - - ^ - ' - ^ - - - - - 121 



i 



PRACTICAL GEOMETRY. 



GEOMETRY is the science of extension and magnitude ; it teaches the construction of 
all right-lined and curvilineal figures, and is divided into Theoretical and Practical. 

The Theoretical part is founded upon reason and self- evidence : it demonstrates the con- 
struction of variously formed figures, and evinces the truth, or detects the falsehood on which 
they are made. This is the foundation of the Practical part ; and without a knowledge of the 
Theory, no invention to any degree certain can be made in the Practice. 

The uses of Geometry are not confined to Carpentry and Architecture, but, in the various 
branches of the Mathematics, it opens and discovers to us their secrets. It teaches us to con- 
template truths, to trace the chain of them, subtile and almost imperceptible as it frequently is, 
and to follow them to the utmost extent. 

Its uses are great and necessary in Astronomy and Geography. The science of Perspec- 
tive is entirely dependent upon its principles. To enumerate its many uses is beyond my 
power. Those who desire to become thoroughly acquainted with Geometry, will do well to 
study attentively the elements of Euclid. 

\ 

As my labours are not intended for the abstruse Mathematician, but for the instruction of 
the Practical Carpenter, I shall omit all speculative demonstrations, the sections of Cylinders 
and Globes excepted (which are not to be found in Euclid,) and confine myself to the useful 
part of the science, viz. Practical Geometry. 

[2] 



10 PRACTICAL GEOMETRY. 



PLATE I. 



DEFINITIONS. 

1. ^ POIJVT has position hut not magnitude. 

2. ^ line is length without breadth or thickness. 

3. A superficies hath length and breadth only. 

4. A solid is a figure of three dimensions, having length, breadth, and thickness. 
Hence surfaces are the extremities of solids, and lines the extremities of surfaces, and 

points the extremities of lines. 

5. Lines are either right, curved, or mixed of these two. 

6. A right or straight line lies all in the same direction between its extremities and is the 
shortest distance between two points, as A. 

7. A curve continually changes its directions between the extreme points, as C. 

8. Li72es are either parallel, oblique, perpendicular, or tangential. 

9. Parallel lines are always at the same distance, and tvill never meet though ever so far 
produced, as D and E. 

10. Oblique right lines change their distance, and would meet, if produced, as F. 

11. One line is perpendicular to another when it inclines no more to one side than another, 
OS G. 

12. A straight line is a tangent to a curve when it is produced and touches it without 
cutting, as H. 

13. An angle is the inclination of two lines towards one another in the same plane, meeting 
in a point, asl. - 

14. Angles are either right, acute, or oblique, as K. 

15. A right angle is that which is made by one line perpendicular to another, or ivhen the 
angles on each side are equal, as G. 

16. An acute angle is less than a right angle, as K. 

17. An obtuse angle is greater than a right angle, as L. 

18. A superficies is either plane or curved. 

19. A plane, or plane surface, is that to which a right line will every way coincide; — but 
if not, it is curved. 

20. Plane figures are bounded either by right lines or curves. 






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PRACTICAL GEOMETRY. 11 

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21. £ solid is said to be cut by a plane when it is cut through in any particular place ^ and 
the place that is cut is called the section of the solid. 

22. Plane figures, bounded by right lines, ham names according to the number of their 
sides, or of their angles, for they have as many sides as angles — the least number is three. 

23. ^n equilateral triangle is that whose, three sides are equal, as M. 

24. An isosceles triangle has only two sides equal, as N. 

25. A scalene triangle has all sides unequcd, as 0. 

26. A right-angled triangle has one right angle, asV . 

27. Other triangles are oblique angled, and are either obtuse or acute. 

28. An acute-angled triangle has all its angles acute, as M or N. 

29. An obtuse-angled triangle has one obtuse angle, as O. 

' 30. A figure of four sides and angles is called a quadrangle, or quadrilateral , as Q, R^ S; 
T, U, and V. 

31. A parallelogram is a quadrilateral, which has both pairs of its opposite sides parallel ^ 
cwQ, R; U, and^ ; and takes the following particular names. 

32. A rectangle is a parallelogram having all its angles right ones, as Q and R. 

33. A square is an equilateral rectangle, having all its sides equal, and all its angles right 
ones, as Q. 

34. A rhombus is an equilateral parallelogram, whose angles are oblique, as U. 

35. A rhomboid is an oblique-angled parallelogram, as V. 

36. A trapezium is a quadrilateral which has neither pair of its sides parallel, as T. 

37. A trapezoid hath only one pair of its opposite sides parallel, as S. 

38. Plane figures having more than four sides, are in general called poly gons, and receive 
other particular names according to the number of their sides or angles. 

39. A pentagon is a polygon of five sides, a hexagon has six sides, a heptagon seven, an 
octagon eight, a nonagon nine, a decagon ten, an undecagon eleven, and a dadecagon twelve 
sides. 

40. A regular polygon has all its sides and its angles equal ; and if they are not equal, the 
polygon is irregular. 

41. An equilateral triangle is also a regular figure of three sides, and a square is one of 
four ; the former being called a trigon, and the latter a tetragon. 

42. A circle is a plane figure bounded by a curve line called the circumference, which is- 
every where equidistant from a certain point within, called its centre. 

' 2* 



12 PRACTICAL GEOMETRY. 

43. The radius of a circle is a right line drawn from the centre to the circumferencej as 
a b ai W. 

44. A diameter of a circle is a right line dravm through the centre^ terminating on both 
sides of the circumference ^ as c dat W. 

45. ,^n arch of a circle is any part of the circumference, 

46. A chord is a right line joining the extremities of an arch, as a. h at X. 

47. A segment is any part of a circle bounded by an arch and its chord, as X. 

48. A semicircle is half the circle^ or a segment cut off by the diameter, as Y. 

49. Jl sector is any part of a circle bounded by an arch and two radii, drawn to its extre- 
mities, as Z. 

50. A quadrant, or quarter of a circle, is a sector having a quarter of the circumference for 
its arch, and the two radii are perpendicular to each other, as Al. 

51. The height or altitude of any figure is a perpendicular let fall from an angle, or its 
vertex, to the opposite side, called the base, as a. h at B 2. 

52. When an angle is denoted by three letters, the middle one is the place of the angle, and 
the other two denote the sides containing that angle ; thus leth\ic be the angle a^ C 3, b is the 
angular point, and a b, and b c are the two sides containing that angle. 

53. The measure of any right-lined angle is an arch of any circle contained between the two 
lines which form the angle, and the angular point being in the centre, as D 4. 

PLATE II. 

PROBLEMS. 

Figure 1 . To draw a Perpendicular to a given Point in a Line. 

A B is a line, and c a given point ; take a and b, two equal distances on each side of c, 
and with the foot of the compasses in a and b make an intersection df and draw d c, which is 
the perpendicular. * 

Fig. 2. To make a Perpendicular with a Ten Foot Rod. 

Let a 6 be six feet, then take eight feet, and from a make an arch at c b, and from the point 
a with the distance of ten feet cross at c, then draw c b, which is the perpendicular. 



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PRACTICAL GEOMETRY. , is 

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Fig. 3. To let fall a Perpendicular from a given Point to a Line. 

In the given point c make an arch to cross the line in a and bf and from a and b make an 
intersection at d, and draw e d the perpendicular. 

Fig. 4>. To draw a Perpendicular upon the End of a Line. 

Take any point d at pleasure above the line, and with the distance d b make an arch ab e, 
and draw a line a c? to cut it at c, and draw c b the perpendicular. 

Fig. 5. To divide a Line into two equal Parts by a Perpendicular. 

From the extreme points a and b describe two arches to intersect at c and d, draw c d, 
which divides the line into two equal parts. 

Fig. 6. To divide any given Angle into two equal Angles. 

Take two equal distances a b and a e on each side of the angular point a, and with the same 
opening of the compasses or any other, place the foot in b and e, make an intersection at d, and 
draw d a, which will divide the angle into two equal parts. 

Fig. 7 and 8. An Angle being given, to make another equal to it, from a given Point in a 

right Line. 

Let 6 a c be the angle given, and c dz. right line, c the given point, on a make an arch b c 
with any radius, and on c with the same radius describe an arch d e, take the chord of b c, set it 
from d to e, and draw e c, then the angle e cd will be equal \.o c ab. 

PLATE III. 

Fig. 1. Upon a right Line to make an equilateral Triangle. 
Take a b the given side from a and b, and make an intersection at c, and draw c a and c b. 

Fig. 2. Upon a right Line to make a Square. 

With the given side a b, and in the points a and b, describe two arches to intersect at e, di- 
vide b e into two equal parts at/, make e d and e c each equal to «/, draw ad, d c, and c b. 



r 



14 PRACTICAL GEOMETRY. 

Fig. 4, 5, 6. The side of any Polygon being given, to describe, the Polygon to any JVumher 

of sides whatever. 

On one extreme of the given side make a semicircle of any radius, but it will be most con- 
venient to make it equal to the side of the polygon ; then divide the semicircle into the same 
number of equal parts as you would have sides in the polygon, and draw lines from the centre 
through the divisions in the semicircle, always omitting the two last, and run the given side 
round each way upon these lines, join each side, and it will be completed. 

Example in a Pentagon. Fig. 4. 

Let abhe the given side, and continue it out to c; on a, as the centre with the radius a b, 
describe a semicircle, divide it into five equal parts; through 2, 3, 4, draw a 2, ad, ae; make 
b e equal to a 6, 2d equal to % a or ab ; join 2d, de, and e b. In the same manner may any 
other polygon be described. 

N. B. This depends upon the equality of the angles upon equal arcs. See Fig. 3. 

Fig. 7. Through a given Point a, to draw a Tangent to a given Circle. 

Draw « o to the centre, then, through a draw b c perpendicular to a o, it will be the 
tangent. 

Fig. 8. ^ tangent Line being given, to find the Point wJiere it touches the Circle. 

From any point a in the tangent line b a, draw a line to the centre o, and divide a o into two 
equal parts at m and with a radius m a, or m o, describe an arch, cutting the given circle in n, 
which is the point required. 

Fig. 9. Two right Lines being given, to find a mean Proportion. '< 

Join a b and b c in one straight line, divide a c into two equal parts at the point o, with the 
radius a or o c describe a semicircle, and erect the perpendicular b d, then is a b to bd a.sb d 
is to 6 c. 

Fig. 10. Through any three Points to describe the Circumference of a Circle. 

From the middle point b draw chords b a and 6 c to the two other points a and c, divide 
the chords a b and b c into two equal parts by perpendiculars meeting at 0, which will be the 
centre. 



FLy/e ,/. 




PRACTICAL GEOMETRY. 15 

To find the length of any Arc A B C ©/"a Circle, 

Draw the chord Ji C and produce it to E ; bisect the arc AB C inB, aud make A D 
equal to twice A B ; divide C B into three equal parts, and set one out to jE ; then AE'is the 
length of the arc. ' 

PLATE IV. 

Fig. 1. Three Lines being given, to form a Triangle. 

Take one of the given sides a b, and make it the base of the triangle ; take the other 
side a c, and from a, describe an arch at c ; then take the third side b c, and from b, de- 
scribe another arch crossing the former at c, and join a c and b c. 

JVote : that any two lines must be greater than a third. 

Fig. 2, 3. To make a Quadrangle equal to a given Quadrangle. 

Divide the given quadrangle, ^^. 2, in two triangles ; make the triangle efg equal to a be, 
and e g h equal to a c d, and it is done. ^ 

Fig. 4, 5 . Any irregular Polygon being given, to make another of the same Bimensions. 

Divide the given polygon, fig. 4, into triangles, and in fig. 5, make triangles in the same 
position, respectively equal to those in j^^. 4; then will the irregular polygon/, ^, A, z, A, be 
equal and similar to ah cde. 

Fig. 6. To make a Rectangle equal to a given Triangle. 

Draw a perpendicular c d, divide it into two equal parts at e, through e draw fg, parallel 
to the base a b, draw af, b g, perpendicular ; then will the rectangle a b gf be equal to the 
triangle a be. 

Fig. 7. To make a Square equal to a given Rectangle. 

Let tt 6 c (/ be the given rectangle ; continue one of its sides as a 6 out to e, make b e equal 
to the other side be, divide a e into. two equal parts at i, with the radius ie or ia make a 
semicircle afe, and draw bf perpendicular to a b ; make the square bfgh, which is 
equal to the parallelogram abed. ^ 



<«^*- 



14 PRACTICAL GEOMETRY. 

Fig. 8, To make a Square equal to two given Squares, 

Make the perpendicular sides a c and a b of the right-angled triangle cab equal to the sides 
of the given squares t^ and Bf draw the hypothenuse c b, which is the side of the square D, 
equal to the squares A, B, C. 

« 

PLATE V. 

Fig. 1. To draw a Segment of a Circle to any Length and Height. 

a bis the length, i h the height; divide the length a b into two parts by a perpendicular g 
c : divide h by the same method, then their meeting at g will be the centre ; fix the foot of the 
compasses in g, extend the other leg to A, make the arch a h b, which is the segment. 

Fig. 2. To draw a Segment by Rods, to any Length and Height. 

( Make two rods c e and c/ to form an angle ecf, so that each may be equal to a b, the open- 
ing ; place the angle c to the height, and the edges to a and 6, put a piece a b across them to 
keep them tight, then move your lath^round the points «, b, and it will describe the segment at 
the point c. '% 

Fig. 3. To describe a Segment of a Circle at twice, upon true Priiiciples, by aflat Triangle. 

Let the extent of the segment be a b, its height c d, from the extreme b to the top d draw 
b d, through the point d draw e d parallel to the base a b, equal in length to d b, describe one 
half, as you see at Gj then move your nail, or pin, out of a, stick it in the point b, and describe 
the other half. 

Fig. 4. The transverse and conjugate Axis of an Ellipsis being given, to draw its Re- 
presentation. , 

Draw a d parallel and equal to n c, bisect it in c; draw e c and d g cutting each other at 
m, join m c, bisect it by a perpendicular meeting c g, produced at h ; A.vsmhd, cutting b aaXk, 
and make n i equal to n k ; nl equal to n h; through the points i, I, k, h, draw the lines h i, 
k I, and ^ I, h k, then describe the four sectors by help of the centres, i, I, k, h, and it will be 
the representation required. 



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PRACTICAL GEOMETRY. ir 

Fig. 5, To describe an Ellipsis hy Ordinates. 

Make a semicircle on the length a b, divide it into any number of equal parts, as 16, on 
the end at a make a 8 perpendicular, equal to half the width, and draw the ordinates through 
all the points in the semicircle, draw the line 8, 1 to the centre, then a 1 8 will be a scale to set 
your oval off; take 1 1 from the sca;le, and set it from 1 to 1 in your oval both ways at each 
end ; then take 1 2 in your scale, and set it to 1 2 in the oval, and find all the other points in 
the same manner ; a curve being traced through these points will be the true ellipsis. 

PLATE VI. 

Fig. 1. To make an Ellipsis with a String. 

Take the half a ^ of the longest diameter a h, and with that distance fix the foot of the com- 
pass in c, cross a 6 at e, /, in which stick two nails or brads, then lay a string round ef and c, 
fix a pencil at c, and move your hand round, keeping the string tight, will describe the 
ellipsis. 



te^Vf^u 



Fig. 2. To describe an Ellipsis by a Trammel. 

1 2 3 is a trammel rod : at 1 is a nut with a hole to hold a pencil ; ^^^H 3 are two 
other sliding nuts ; make the distance of 2 from 1, half the shortest diamete^Ji^ur ellipsis, 
and from the nut 1 to 3 equal to half the longest, the points 2 and 3 being put into the 
grooves of the same size, move your pencil round at 1, and it will describe the true curve 
of an ellipsis. 

Fig. 3. t^n Ellipsis being given, to find the Centre and two Axes. 

Draw any two parallel lines a b and c c? at pleasure, divide each of them in two equal parts 
at the points e and/, and through ef, draw the line k I, divide k /into two equal parts at the point 
g, place the foot of the compass in g, with the other foot make two crosses h and i, on the cir- 
cumference ; draw a line h i, through g, draw m n parallel to h i, also through g draw op, at 
right angles to m w; then o p\% the transverse axis, and m n the conjugate, and g the centre 
of an ellipsis. 

[3] '--.. 



18 PRACTICAL GEOMETRY. 

Fig. 4. How to proportionate one Ellipsis within another ; that is, to give it the same Length 
in Proportion to its Width, as the Length of the other has to its Width. 

Let the given ellipsis he adb c, make the parallelogram e hf g to touch the sides and ends 
of the ellipsis, draw the diagonals ef, and g h, of the rectangle, let rqhe the width of the lesser 
ellipsis given, through the point q, or r, draw lo, or m n, parallel to the transverse axis, at the 
points m and n, where it cuts the diagonal, draw m /and n o parallel to the conjugate axis, 
will also show its length. . 

Fig. 5. How to describe an Ellipsis about a Parallelogram, to have the same Length in Pro- 
portion to its Width, as the Length of the Parallelogram has to its Width. 

Let the given parallelogram heab c d; let the diagonals a c, and b d, he drawn from the 
centre i ; draw the quarter of a circle, 2 1 A, to half the width of a rectangle ; divide the quad- 
rant into two equal parts at 1 ; through the point 1, draw the line 13 parallel to the transverse 
axis t6 cut the diagonal bd in the point, 3; then draw the lines 32 and34; again, draw/c? 
parallel to 2 3, then if will he half the width, and de parallel to 3 4 ; and i e will be half the 
length of the ellipsis : make i h equal to i e, and i g equal to if, which will give the four points 
through which the ellipsis must pass ; describe the curve, and the thing will be done. 



JEig. 6. To (^vide a Line in the same Proportion as another is divided. 

d a i^^^^Kiven already divided, and c? e is a line to be divided in the same proportion, 
making an^m^e at c? join ae, draw bf and eg parallel to ae; then deis divided at/ and g 
in the ratio of ad atb and c. 



Fig. 7. To do the same by an equilateral Triangle. 

a 6 is the given line divided : from c take two equal distances ' cd, ed, and by drawing 
lines from the several points in « 6 to c cutting dd, dd will be divided as a 6. 

Fig. 8. To make an Octagon the nearest Way from a Square. 

Draw the diagonals of the square to cross at e, fix the foot of your compass in c, and take 
the distance ce and make an arch/ eg', then set your gauge to df or b g, which will gauge 
oiF each angle. 



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PRACTICAL GEOMETRY. 19 



♦ PLATE VII. 

come SECTIOJyS by mTERSECTmG LUVES. 

DEFINITIONS. 

1 . ,A cone is a solid having a circular base, from which the sides continually diminish in straight 

lilies to a point in which they all terminate, and this point is called a vertex, 

2. Opposite cone is another cone joining the vertex of the other cone, with its sides every 

where in the same straight line passing through the vertex as a common point. 

3. A right line joining the vertex and the centre of the base is called the axis. 

4. If a cone be cut by a plane passing entirely through its curved surface, but not parallel to 

the base nor to the axis, nor to a plane touching the side of the cone, the section is an el- 
lipsis, excepting in one position which is a circle. 

5. If a cone be cut by a plane parallel to its sides, the section is a parabola. 

6. j^ the cone be cut by a plane passing through the opposite cone, the figure will be a hyperbola. 



To describe the Ellipsis from the Cone. 

FiGunE A. Let B be half the circle of the base of the cone, n the vertex at the top ; 
then n a and n d are two sides ; let the cone be cut by a plane passing through g h ; bisect g h 
at the point k, and through k draw r q, parallel to the base a d ; also, bisect r qmm, describe 
the semicircle r p q, draw kp dX right angles io qr; and ^ ^ is the length of the ellipsis, and 
h k half its width ; from which the figure may be described at C, as explained in the next 
plate. 

To describe the Parabola from the Cone. 

Figure A. Let ie be the axis of the parabola, parallel to the other side ndoi the cone, 
and through e draw e c at right angles to the base ; then will e c be half the width of the para- 
bola, and e I its height ; then the ^^z<re will be described, as at Z), by intersecting lines upon 
each ordinate, up to the crown, from the equal divisions on each side. 

3* 



20 PRACTICAL GEOMETRY. 

To describe the Hyperbola from the Cone. 

f 
Figure A. Let the axis of the hyperbola be if, cut by a plane passing through / and i, 

till it cut the opposite cone at /; draw/ 6 at right angles to ad, then is /^ the height of a 

hyperbola, and/6 half the width of the base, and z/ its 'transverse axis; then make/ i at E 

equal to fi in figure A, make il in jE equal to i I in figure *Q., 6 6 in E equal to twice / 6 in 

figure A; let the base 6 6 in ^ be divided into ten equal parts, as at 1 2 3 4 5, that is, into 

five equal parts on each side from the centre, and draw lines to thie point I through these 

points ; likewise divide the height into five each way, and draw lines to the vertex at i ; this 

will show the points through which the curve must pass. 

PLATE VIII. 

How to draw any Semi- ellipsis upon the transverse or conjugate Axis, or even a Semicircle it- 
self, by a new Method of intersecting Lines, 

Figures A and B. Let the given axis be ab, and let it be divided into any number of 
parts, as 10 ; also let the height be divided into half the number of parts ; make e d equal e c, 
that is, to the height of the arch ; then, from the point d, draw lines through the equal divi- 
sions of the axis a b ; likewise, through the points 1, 2, 3, 4, 5, in the height af, draw lines 
tending to the vertex at c, which will intersect at the points h, i, h, I ; and lines being drawn 
through the divisions of bgto c, at the crown in the same manner, will give the points n, o, p, 
q ; a curve being traced through these points, will form the true curve of an ellipsis. 

The semicircle, figure C, is drawn in the same manner, by making a f equal to one half 
oi ab. 

How to draw the true Segment of a Circle, by the Method of intersecting Lines. 

Figure D. Let a 6 be the length of the segment, and o c its height, and draw the chord 
b c for one half of the segment, and draw bm a.t right angles to 6 c ; and from the point o di- 
vide a b each way, into five equal parts; also from c, divide cm, and en, each into five equal 
parts ; and draw 1 1, 2 2, 3 3, 4 4, 5 5, on each side, through the divisions 1, 2, 3, 4, 3 on as, 
and 1, 2, 3, 4, 5 on 6 r ; draw lines to c, which will intersect the other lines at the points d, e, 
/, g, and h, i, k, I: the curve being traced, the thing is done. 



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PRACTICAL GEOMETRY. si 



Hoiu to draw a flat Segment of a Circle nearly true. 

Divide the length of the segment into equal parts each way, from the centre d, as before, jl 

and draw the lines 1 1, 22, 3 3, 4 4, 5 5, all at right angles, to the length «6; lines being ' 

drawn to the crown at c, from the divisions at each end, will show the points which the segment . 
must pass through ; the curve being traced, the thing is done. -"f^ ' - 

Remark. Although this last method is not the true segment of a circle, but a parabolic 
curve, yet it will be found useful in practice, in tracing any segment whose height is not more 
than one tenth part of its length ; if the centre of the segment is found, and drawn with a com- 
pass, the diiference will hardly be visible, and the flatter the segment, this difference will be- 
come the more imperceptible ; but if the height exceeds one tenth of its length, the difference 
will be visible ; for then the arch will be quicker at the vertex, and get flatter and flatter to- 
wards each extreme. 

In the same manner may all kinds of rampant ellipses be described, or any segment of them, 
as at F and (r, also a rampant parabola in the same manner, as H. 



PLATE IX. 

THE SECTIONS OF A CYLINDER. 

DEFINITIONS. 

tS. cylinder is afl,gure generated hy the revolution of a right-angled parallelogram about one 
of its sides ; consequently the ends of the cylinder are equal circles, and the line passing 
through the centre of the cylinder is called the axis. 

The section of a cylinder, cut by any plane, is an ellipsis, or circle, or rectangle, proved by the 
writers on Conic Sections. 



£2 PRACTICAL GEOMETRY. 



To find the Section of a Semicylinder, by Ordinates, when it is cut at right Angles to the 
Plane passing through its Axis, in the Direction ab. Fig. 1. 

Let the circle of the base be divided into equal parts at A, and drawn parallel up the cylin- 
der to the line ab, at the points 0, 1, 2, 3, 4, 5, &c. and from these points draw lines at right 
angles to a 6 ; then B being pricked from A, as the figures direct, B will be the section of the 
cylinder. 

DEMONSTRATION. 

Conceive the semicircle A at the base to be turned at right angles to the plane, also the semiellipsis B at right 
angles to the same plane ; then will the ordinates of JB be parallel and perpendicular over the ordinates of A, and 
eveiy corresponding point in the circumference of jB will fall perpendicular to the same corresponding points in A : 
therefore B is the true section of the cylinder, cut in this position. 



To cut a Cylinder in the Direction a b, upon a Plane, passing through its Axis, to make an 

acute Angle with the Plane. Fig. 2. 

Let C, at No. 1, be the given angle, which the section at B is to make with the plane of the 
cylinder; take ab in figure 2, that is, the radius of the base, and set it from 6c, at No. 1, 
perpendicular to z 6 ; draw c c parallel to i b, also from c draw c e perpendicular to i b; then 
take the distance ci, set it from i tof, in figure 2, a.t B ; likewise take i e from No. 1, and set 
it from i to e va. figure 2, a,t B ; draw ed parallel to m n, to cut the rake in d, and join df; 
then is df the bevel of the first ordinate of the section B. And draw the lines ec and da 
parallel to the axis ; join a c at i^ ; then will a c be the bevel of the first ordinate of the base. 
Draw all the other ordinates of A parallel to a c, and at the points 1, 2, 3, 4, &c. in m n, 
draw lines parallel to the axis of the cylinder, to cut the raking line VTVai 1, 2, 3, 4, 5, &c. 
From these points let lines be drawn parallel to df; then the ordinates of B, being pricked 
from the same corresponding ordinates of the base at A, will give the section of the cylinder. 

JVbte. The point/ will fall beyond the sweep at the section B. 



PRACTICAL GEOMETRY. 



DEMONSTRATION. 



Let the plane B be conceived to be turned round the line V Wto make an angle at the point i, with the plane 
nm VW equal to the angle e i c, No. 1 ; and conceive a straight line drawn from e perpendicular to the plane 
nmVfV, the line thus supposed to be drawn will be parallel to the plane <3. of the base, and the triangle formed 
by if, i c, and the perpendicular from e, will be equal and similar to the triangle cie, No. 1 : then because d e and 
the perpendicular are both parallel to the base, thfe line that joins the points e and /will also be parallel to the 
base ; and because e d is equal to 06, the triangle e d/ will be equal and similar to the triangle 6 a c in the plane of 
the base .A : and because 6 c and the perpendicular drawn from e are both in a plane parallel to the axis, the plane 
passing through d/and ac will also be parallel to the axis: but df is also" in the plane of the section, for the point 
d is the intersection VW, and the point/will be in the! perpendicular drawn from e ; therefore, if.a series of planes 
be conceived to be drawn through the ordinates of the base parallel to the plane passing through a c and df, the in- 
tersection of these planes on the plane of section B will be parallel to the ordinates of jB, and every two correspond- 
ing lines will be in a plane parallel to the axis, and therefore as the lines formed by the intersections of the series 
of planes in the section B, and equal to those in the base A, the extremities are in the section of the cylinder. 



To cut a Segment of a Cylinder, in the Direction a b, to make an obtuse tdngle with the 

Plane of the Segment. Fig. 3. 

Let No. 1 be the angle given, which the section B is to make with the plane of the seg- 
ment; from/ in No. 1, draw fgaX right angles to fc, and g e also perpendicular, to make 
the right-angled triangle egf. And in figure 3, at B, draw gf, at right angles to ab, and 
make g e equal to ^g- e at No. 1. Also, make gf at B equal to gf at No. 1. Draw e c? at 
B, parallel to MJV, and at the point d, where it intersects the line a b, join df', then df'is, one 
of the. ordinates. From e and d, draw the two parallel lines ec and (/3, join c3 ; then e3 
will also be an ordinate of the base. Draw parallel lines at discretion to c 3, for the other 
ordinates of the base ; and from their intersection upon m n draw lines parallel upon the cy- 
linder, to cut a 6 in 1, 2, 3, 4, &c. and from these points draw parallel lines to df, which are 
the ordinates of B ; these? being pricked from the base as the figures direct, will give the 
points through which the curve must pass, which being traced, will be the true section of the 
segment of the cylinder. 



24 PRACTICAL GEOMETRY. 

DEMONSTRATION. 

Is the same as the preceding Demonstration. 

That the reader may perceive this more clearly, the best way is to draw those lines on pasteboard. 
The section and the end being made to turn round,, in their proper position, then the demonstration will be clearly 
seen. 

Figure 4 is to be laid down and demonstrated in the same manner as Figure 2. 

Remark. Upon these figures depend the whole principles of hand-rails for stairs. The reader*ought to un- 
derstand how to form the section of a cylinder, in any case whatever ; for the face of raking mould of a hand-rail 
is nothing but the double section of a cylinder, as in figure 4, at B, where the double circle upon the base Ji repre- 
sents the plan of a rail, and the bevel at No. 1, figure 4, represents the spring of the plank, and a b the pitch of the 
rail : therefore, it is very necessary that the reader should have a knowledge of these figures and their demonstra- 
tions ; and not be satisfied with only domg it, but read these demonstrations, and consider them with attention : 
then he will be able to see the reason why every line is drawn in the manner it is. 

PLATE X. 

THE SECTIOJVS OF A GLOBE, OR JIJ^Y OTHER FIGURE STAJVDmG UPOJV Jl CIRCU- 
LAR BASE; ALSO, IHE SECTION OF AJ^Y FIGURE STAJVDIJVG Ojy AJV IRREGULAR 
BASE. 

DEFINITION. 

A globe is a figure generated by the revolution of a semicircle round its diameter ^ which be- 
comes the axis of the globe. 



AXIOMS: OR, SELF-EVIDENT TRUTHS. 

1st. Froin this definition it appears^ that every two sections passing through the centre, are 
equal to each other. 



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PRACTICAL GEOMETRY. 25 

2d. Every section of a globe, cut by a plane, is a circle ; for the generating circle may be made 
to revolve round any line, as an axis; and therefore every point in it will generate a circle, 
whose diameter must be twice the radius of that circle distant from the axis of the globe. 

3d. If a semi-globe is cut by a plane at right angles to the plane of its base, the section will be 
a semicircle. 



To find the Section of a Semi-globe cut by a Plane at right .Angles to the Plane of its Base. 

Fig. 1. 

» 

It appears from the last axiom, that there is no tracing required : for, let the section be cut 
across a b, figure 1 ; divide a b in two equal parts at the point c ; and on c, as a centre with 
the radius caov cb, describe the semicircle i^, which is the true section required. 

The same Ordinates. Fig. 2. 

Draw any line de through its centre, and let a 6 be the place of the section upon the base, 
as before ; place the foot of your compass in the centre of the globe at/, and, with a radius t c, 
draw an arch from c to g, in the diameter de ; the foot of your compass remaining still in/, 
draw the concentric dotted circles from c b tofe, and at the intersecting points 1 3 3 4 5 in 
fc, and likewise in c b, erect perpendiculars to those lines j then ^ being pricked from C, as 
the figures direct, will give the points through which this semicircle must pass. 



DEMONSTRATION. 

Conceive the semicircle C to stand at right angles upon d e, also the section A to be at right angles to a 6 ; 
now it is evident if g t is the height of the globe over the point g in the base, c 1, which is equal to g 1, must also 
be the height of the section, because the points c and g stand at an equal distance from the centre ; and therefore 
the point 1 over c, is in the surface of the globe. In the same manner it may be proved, that any other points car- 

[4] 



26 PRACTICAL GEOMETRY. 

ried round by the dotted lines are in the same surface ; but the section that stands upon a b, in A, is a semicircle ; 
and consequently the method of tracing is also a semicircle. 

Observation. Hence appears the erroneous principle of tracing used by a late writer upon this subject, as you 
may see at figure 2, where A is the section of a globe, and the bracket at D is the section across the diameter. A 
is truly traced from 2?, because the ordinates are carried round in circles; but by his method of tracing, as you 
see at C, upon the other side, the point of the bracket C falls within the sweep of the circle, by reason of the 
ordinate of C being carried straight through between the two bases, which I have proved to be false. And this he 
has applied in bracketing up the angles in the square well hole of a stair case, to the circular curb of -a sky-light, 
which if truly done, is nothing else but upon the same principle as the sections of a globe. 

Figure 3, is done upon the same principle as figure 1. ./2 is the section traced from 
C, and wants no other demonstration than what has been given in figure 1. 

Figure 4, is an ogee section, standing upon a circular base across the diameter; and 
A is the section traced from it, upon the same principles a.s figure 1. 

From these examples it is clear that this method of tracing does not depend on the form 
of the top, but entirely upon the base. These figures are supposed to be generated round 
an axis ; and, as every circle is carried round at an equal distance from the axis, the per- 
pendicular height of the figure, upon any circle, must be the same height in every point through- 
out the circle: which proves itself to^be the only method for any thing of this kind. 



»5 Semi-globe being cut by a cylindrical Surface perpendicular to the Plane of it^ Base, 
to find the Form of a Veneer that will bend round it. Fig. 5. 

Let de he drawn through the centre/; and place the foot of your compass in/, the 
centre; and draw the points 6, 1, 2, 3, 4, which are equally divided from the centre at b, in 
the circular surface, draw the concentric dotted lines round to the diameter c? e, at 0, 1, 2, 3, 4, 
and at these points raise the perpendiculars 00, 1 1, 22, 3 3, 4 4. Take the stretch-out round 
612 345, which is one half; and lay it upon the base of No. 1, each way, from 01234, &c. 
and No. 1 being pricked from A, figure 5, as the figures direct, will give the points through 
which the curve must pass for the veneer. 



PRACTICAL GEOMETRY. Z7 



DEMONSTRATION. 



For, since the section standing upon d e is a semicircle, which is equal to the semicircle upon the base; 
and as the points 1 ^ 3 4 in the circular surface, stand at the same distance from the centre /, as 0, 1, 2, 3, 4, in 
de ; now if the point o at No. 1, is made to coincide with the point b in figure 5, then the height o o, standing over 
the point b, will be equal to the height o o at .5 ; but these points are at an equal distance from the centre, therefore 
the top of each ordinate will be in the surface of the globe. In the same manner every other point may be prov- 
ed, when bent round and elevated, to be of the same height, and at an equal distance from the centre with those of 
j3 ; and therefore No. 1, is the true form of the veneer. 

Thjind the Bibs of a Gothic JViche, being the Plan, and JVb. 1, the Front Elevation. 

FIGURE 0. 

Take the length of each base upon the plan, and make them the bases of No. 2, No. 3, 
No. 4, and No. 5 ; divide each base into five equal parts ; also divide the half of No. 1, into 
six parts, and draw the ordinates from the equal divisions, perpendicular to each base ; then 
prick each from No. 1, as the figures direct, will give the form of each rib. This wants no 
demonstration. 



4* 



riale 11. 

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V 



OF CARPENTRY. 



LININGS FOR SOFFITS. 

DEFINITION. 

The linmg of a Soffit in the Theory of Carpentry, signifies the covering of any cbncave Sur- 
face of a Solid spread out on a Plane if possible. 

Soffit in tirchitecture is the under side of the head of a door, window^ or the intrados of an 

archj and may be either plane or curved. 



PLATE IX. 

Hoio to stretch out a Soffit, when a Window or Boar, having a semicircular Head, cuts into 

a straight Wall, in an oblique Direction. 

LET C be the plan or opening of the window, in fig. A, and let the base of the semicircle 
B be drawn at right angles to the jambs, or sides of the plan C ; divide the semicircle into 
any number of equal parts, as ten, and draw the ordinates across the plan, extend the parts 
round B upon the stretch-out line, the ordinates being drawn from the divisions across, and 
traced off from the plan C, as the figures and letters direct, the lining of the soffit will then be 
completed. 

If you would make a cylinder to be only the thickness of the wall, i? shows the end^ of it. 
which is to be traced from the semicircle B. 






30 LININGS FOR SOFFITS. 

How to draw the lining of a Soffit when the Top is a Semicircle^ cutting right into a cir- 
cular Wall. 

Fig. E. This aiid the other below are performed the same as that above, with this differ- 
ence, that you are to prick from the circular plan, instead of the straight plan. 

Fig. I, shows the method when a circular headed window cuts oblique into a circular 
wall. 

JVbte» In all kinds of cylindro-cylindric soffits, when the two jambs are parallel, the 
straight line, which the soffit is pricked from, must be drawn at right angles to the jambs, 
as is shown in this plate ; for want of this consideration, they are shown in books upon wrong 
principles. 

But in the following soffits, where the jambs are not parallel, they must be continued till 
they meet in a point, and the line which the soffit is to be pricked from, must be made to form 
an isosceles triangle with the jambs. 



PLATE XII. 

To draw the lining of a Soffit in a straight Wall; splaying equally all round with a cir- 
cular Head. 

In fig. Jl, continue the sides of the plan .^, that is, a c and b d, to meet at e ; then about 
the centre e, and from the points a and c, describe the soffit O, and stretch the semicircle B 
along the outline of the soffit C, it will be completed. 

To draw the lining of a Soffit in a circular Wall, splaying equally all round with a 

circular Head. 

Fig," B. The stretch-out of this soffit is managed the same as in the last ; draw the or- 

,. t. s of the semicircle B, from thence continue them to /, the concourse of the splay, 
dinatei. -if 

d 1 1^ xe points a, 6, c, d, e, where they intersect the plan, draw the parallel hnes, a e, 6/, eg, 



Liiuiuj for 
ii soffit aiftinii ri<jJit 
into o st7ni(//it wtiJJ 

Flnini] <-t/iui/h- dU Toiin.l %' 



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LININGS FOR SOFFiETS. 31 

•* '■ .» . ■ . " ■. 

&c. parallel to the base of B, and from the points e,/, gf h, and i, circle lines to a, &, c, d and 

e round the centre/, which will give the half of one edge of the soffit, the other half being 

pricked from it ; the other edge is found in the same manner. 

Note. This cannot be pricked from the plan as the others are, as the lines round the splay 
are not level with the plan, and will therefore be longer than those on the plan. 



DEMONSTRATION of Fig. A. 

Conceive the semicircle B to be turned at right angles to the plan Ay then every point in the circumference of 
the semicircle B will be at an equal distance from the point e, but the soffit C is described with the same radius ; 
therefore the edge of the soffit C, that is, the arch line o/, will exactly coincide with the arch of the semicircle 
jB, which was to be proved. 

DEMONSTRATION OF Fig. S, 

It is easy to conceive from the last demonstration, that if the semicircle B is turned up, and the soffit at C 
bent round it, the points 1, 2, 3, 4, 5 at C will coincide with equal divisions in the semicircle JS, and the points 
o 6 c d, &c. at C, will fall perpendicularly over the points abed, &c. in the plan A : for the arches a e, bf, eg, d k, 
and et at C will fall over the parallel straight lines ta,fb,gc,hd,ie, in the plan A, which was to be demon- 
strated. "'■ .,••'''* -3 -*;^^ ♦-":;. 

.:^< :-*.''■■■ ^ 

The learner is advised to cut these and the following soffits outTDT pasteboard, and their 
demonstrations will be more clearly seen. 



LININGS ^OU SOFFITS. 



PLATE XIII. 

To find the lining of an Aperture whose Plan A B C D is a Trapezoid, with two parallel 
Sides A B and D C, which represent the out and insides of the Wall, and two equally/ in- 
dined sides AD and'RC, which represent the Jamhs, and whose elevation A IB, on the 
inside of the Wall is a semi- ellipsis, and that on the outside DGC a Semicircle, so that a 
straight Edge may every where coincide with the lined Surface, and be parallel to the 
Horizon:.^ 

Produce .^ Z) and 5 C to i? ; bisect 2) C at F, and draw EF', produce it to /and it 
will cut AB at H; then FG and HI are equal to each other. Divide the quadrant D G 
into any number of equal parts, (as five,) and draw lines through the points of division cut- 
ting the base D C : from the points of division and in the same straight line with the point E, 
draw lines to cut A H, and the lines so intercepted will represent the level straight lines on the 
soffi-t. Make jEJJ" perpendicular to ^Z) equal to jPG^, and mark the other divisions owEJ 
from j5^, at a, b, c, d, respectively equal to those i^ FD: then take the distances of the seve- 
ral points \n D C from E, beginning next E D, and proceeding to the last E F, and describe 
the arcs from the centres a,b, c, d respectively ; with the fifth part of the arc jD G fix the foot 
of the compass in D, and cross the first arc at c ; place one foot of the compass in e, cross the 
next arc at/, proceed in this manner to i, then D, e,f,g, h, i, will be the coincident line of the 
lining or interior covering for the arc D G. Join i J and produce it to K; make the angle 
KJL equal to the angle KJE, and make JL equal to JE : mark the divisions on JL, so 
that the distances from /may be equal to the distances of the several divisions on JE, then 
the other half of piano- cunioidal line may be found by inversion. Produce the lines a e, bf, 
eg, dhfji, &c. to o,p, q, r, s, &c. make eo,fp, gq, &c. in inverted order equal to the seats 
of the lines on the sofiits and the points o,p, q, r, &c. then curves being drawn through the 
points o,p, q, r, s, &c. and through e,f, g, h, &c. will form the wall lines of the covering, so 
that AD MJVwiW be the whole covering or interior development. 



r/.ih- /■; 




r/af^ 14. 




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1 \ 



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X.. 



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Fu/.F .17V. 





12 3 1 -4 H 2 7 O 1 



LININGS FOR SOFFITS AND TRACERY. S3 



PLATE XIV. 

To find the lining of an Aperture covered with the same Surface, and terminated by a 

circular Wall. 

Find the line D Q M, z& in the last plate, as in a straight wall, then transfer the dis- 
tances from the straight line on the plan to the arcs, representing the faces of the wall to the 
covering, and the edges will be obtained as in the last. 



PLATE XV. 



To draw the lining of a cylindrical Soffit, cutting right in a Wall which does not stand 
perpendicular to the Ground, to a level Base. Fig. A. 

Let a e at i) be the level of the ground, a I the inclination of the wall, equal to the radius 
of the cylinder ; let fall the perpendicular from / to c, in the bottom line a e make the semi- 
circle in fig. A ; to the width of the .cylinder, or the double of a / at D, take the distance a c 
at D, and make b a equal to it in fig. A, and describe a semi- ellipsis to the length of the semi- 
circle d d, and to the width a b ; lay the equal divisions round the semicircle in fig. A in C, 
along the line dd, on each side of the middle point 4, then take the parts ed, d c, cb, b a, 
from the plan B, and lay them at D respectively from e towards d, c, b, and from / draw / e to 
make a right angle with / a, and at the points a, b, c, d erect perpendiculars to a e to cut / e at 
/. g, h and i : take the distances e i, i h, h g, and gf, in B, and lay them on the soffit at C re- 
spectively, from \d, 2 c, 3 b, 4 a, each way, then will the straight line dd in the soffit, when 
bent round, be perpendicular over the elliptic line in the plan B, and the curve line dd cb a, &c. 
d will fall over the points ddcb a in the plan : in the same manner the edge of the soffit may 
be brougli to answer any curve line proposed. 

[5] 



34 CHURCH WORK AND RANGING OF RIBS. 

To draw the Arches of Groins by a new Method, whether right or rampant, so that their 
Arches shall intersect or mitre truly together, from a given Arch of any Form. 

hetfg. E be the given arch of a Gothic fornij draw the chord ac for one half the arch, 
divide it into any number of parts, as four, and through the divisions draw lines from the centre 
e to terminate in the circumference at h, g, I, draw lines from c through hg I to cut the perpen- 
dicular ad a.t b,c,d', and if No. 3 is required to be wider, but the same height as fg. E, draw 
the two chords a c and c h for each side of the arch, divide each into four equal parts, as be- 
fore, and set the parts ah, he, cd, perpendicular on each end of a 6 at No. 2, and from the 
divisions draw lines to the vertex at c, then trace the curve through the points h, g, I, &c. so 
the arch at No. 2 will truly mitre into j^^. E ; in the same manner the rampant curve at No. 3 
will be brought to correspond with ^^. jE^, and No. 2.* 



PLATE XVI. 



As it happens sometimes in church work, that windows go higher than the ceiling line, which 
therefore requires to he hollowed out, so that the light may he thrown down into the hody 
of the church: I shall in this place show the method of making a curh for that purpose. 



To find the Form of the Curb. 
Let A 6 / be the head of the window, figure A, and let it come as high as a h, above the 



* It is hardly possible to find ^ more ready method in practice, because a chalk line will soon strike all the 
radical lines, having only to move it but once from the point e up to c at the crown ; fig. F shows the common 
method by dividing the basis of each into a like number of parts, and transferring the height, as the figures explain, 
at No. 1 and No. 2 ; nothing is more tedious in practice than raising a number of large perpendiculars, and going 
continually from one curve to get the height of another. . • 



rhifi- Id. 

y."i.Ficf.A. 



C eihlU/ h?)-:- 7//J/I71 the t7it7iSVL^JSe 



i ci/i^i 




// i' 



CHURCH WORK AND RANGING OF RIBS. 35 

ceiling ;* and let ab ai No. 1, be the same height, and b c the direction of the light, and a c 
^vill be the length of the curb. Make a c at No. 2, equal to a c at No. 1, and divide it into 
six equal parts ; also divide a b, in figure A, into six equal parts, and let the ordinates be 
drawn as is explained in the figure ; a curve being traced round the points of intersection, will 
give the form of the curb. 

Figures B and C show the method of drawing and backing any elliptic rib with a com- 
pjfes, which is exceedingly handy in drawing, and will be near enough for the representation 
of an elliptic rib on paper, as no other method will be so clean when done ; but for practice, 
a trammel, or intersecting lines is more ready. 



To draw and range, the Ribs by this Method. 

^ In B, let c A be the height, and c b the width j divide the difference into three equal parts, 
and set four such parts on each side of c, to d and d, and make an intersection with the dis- 
tance dd dX e, and draw a line through e and d to i, then d and e are the centres for the inte- 
rior side : suppose the rib is to be backed as much as a 6 upon the bottom, set a b from d to/, 
and from e to g, parallel to the base ; and draw a line through g,f, to k j then g and/ are the 
centres for describing the ranging lines. 

The rib E is traced from D, and a b being set all round on the parallel lines, shows how 
the ranging is found for a drawing on paper : the rib C is described in the same manner. 

The word backing is properly applied to the upper side of any thing in Carpentry or 
Joinery, as the back of a rafter, the back of a rail ; but range applies either to the operation of 
levelling the upper or lower edge, and explains its own meaning, viz. forming the edge so as 
to range with the other edges, whether forming a ceiling or the exterior of a roof. 



* The ceiling is here supposed to be level, which is seldom the case in a church ; but the method will be no- 
thing different if the ceiling line a e were to incline to the horizon in any angle whatever, only observe to make 
cat equal to that angle. 

5* 



36 CENTRINGS TO GROINS. 



# 



PLATE XVII. 



DEFINITION. 

Groins are formed by the intersection of arches or vaults, and the surfaces of their meeting 
may he considered as the sections of cylinders, cylindroids, ^c. 

BRICK GROmS. Description. 

P, P, P, &c. is the plan of the piers which the vault is to stand upon, a b, fig. B, is the 
end opening, which is a given semicircle ; and 6 c is the opening of the side arch, which is 
to come to the same height as the end arch a 6 : fix your centres over the body range, fig, Ji, 
as shown in the section at C, then board them over. In fig, *S. is the manner of fixing the 
jack ribs upon the boards, which likewise shows at C 

To find the Mould for the Jack Ribs. 

Take the opening of your arches in fig. A, that is, a b and a d, and lay them down in 
fig. D, at a 6 and b c, to make a right angle. Divide one half of the given semicircle into 
five parts, and square them across 1, 1, 1, &c. to cut b d and dc, the diagonals in 2, 2, 2, &c. 
and through the points 2, 2, 2, &c. draw lines parallel to 1, 1, 1, &c. the base of E, both 
ways towards Pand G ; stick in nails at 1, 2, 3, 4, 5, in the arc of P, and bend a thin slip of 
wood round them, which mark with a pencil at every nail ; this slip of wood being stretched 
out from d, 1, 2, 3, 4, 5, and squared over to G, will intersect the other lines in small rectan- 
gles : a curve being traced through the diagonals of each recte^ngle, will give a mould to 
set the jack ribs. 

How to fix the Jack Ribs." 

Bend your mould G from a, to the crown at e, in fig. .>^j that will give the edges of 
your boards ; then fix a temporary piece of wood, level upon the crown, in the direction of 
/ /, and let it come the thickness of your boards lower than the crown, and it will give 
the height of your jack ribs, which is a very sure method of placing them. 



Flak' ij. 




Mtu 



Pl^le 18. 




CRADLING TO GROINS. %7 

To find a Mould to cut the Ends of the Boards. 

The rib F is traced to the height of E, or got by a trammel, which will be fully exempli- 
fied in the following plates. Take the parts round F, and lay them out to 1, 2, 3, 4, 5 ; then 
^\vill be got in the same manner as G, which will be a mould to cut the ends of your board 
that goes upon the jack ribs against the body range. 

Fig. 1. Is an easy method of getting the moulds when both arches are the same opening. 

Take half the opening of the arches, whatever they are, and draw a quarter circle, and 
divide it into six ; bend a slip round it to take its parts, then stretch it out upon the base from 
to 6, and square over your points 1, 2, 3, &c. Through the points in the arch draw the lines 
on both sides parallel to 0, 6, the curve being traced as before, gives both moulds of an equal 
and similar form. 

Aof«, The curve F may be drawn in practice with a trammel, independent of the other, and the two moulds 
F and G may be drawn separate, without any connexion of lines, as shall be shown hereafter. 



PLATE XVIII. 



DEFINITIONS. 

Groins are said to be ascending or descending when they are not built upon level ground. 



CENTRmG FOR ASCEjYDIJYG OR DESCEJVDIJYG GROIJVS. 

The Plan and Inclination of any Groin being given, and one of the Body Bibs, as B, also the 
Place of the .Angles upon the Plan, to find the Form of the Side Bibs, so that the Inter- 
section of both Arches will be perpendicularly over the Plan. 

Divide half the circumference of the given rib B, into any number of equal parts, and 
draw them to intersect the angles 5 and from thence let them be returned up to the rib C, 



38 CRADLING TO GROINS. 

upon the side; then C being pricked from the given rib at B, as the letters direct, will 
give the form of the side centre. The same is shown at F, by the method of intersecting 
lines. 



To find the two moulds Jy and "Ea for placing the Jack Ribs^ to bend over the Angles in the 
Body Range, when boarded in, so that they may be perpendicular over the Angles upon 
the Plan. 

At C, draw lines from the points a, b, c, d, e,f, g, &c. where the ordinates of C intersect 
the top of the arch, perpendicular to the rake, and draw the semi- ellipsis, A, to the width of 
the body range ; and to a h, the height of the side centres, perpendicular to the rake ; and 
continue the ordinates of B, up to A, to intersect at 1, 2, 3, 4, 5, 6. Bend a slip round these 
points, and mark them opposite to every point, and stretch it out along k, 1, 2, 3, 4, 5, 6, be- 
tween D and E, and draw lines through these points, at right angles to A 6, to intersect with 
the perpendiculars. Begin at 6, and trace a curve both ways, will give the edges of the two 
moulds for placing the jack ribs. 



. To cut the Jack Ribs to the Rake of the Groins. 

Set the number of the jack ribs upon the arch B, at their proper distances, and take their 
several heights, 'that is, h i, kl, mn, and set them upon the arch G, from a to b, and from a to 
c, and from a to d, draw lines through these points parallel to the' rake, which will show how 
the jack ribs are to be cut, so that they shall range properly with the other raking centres. 



Note. All the body ribs must be ranged according to the rake of the groin ; to do this exactly, the under 
edges of all the ribs must be bevelled according to the rake ; then make a mould as B, or one of the body ribs 
themselves will answer instead of a mould, which being applied to each side of any other rib, keeping the bpttom 
fair with the under edge upon each side, and drawing the curves by the other, it will give the ranging line. 



Flatc J9- 




CRADLING TO GROINS. 59 

PLATE XIX. 

Given the tivo Side Arches of any Groin, and the Inclination; to find the Intersection of the 

Angles upon the Plan. 

Divide half of the body rib B into equal parts, and draw parallel lines to b, c, d, e and/; 
and from the point a, as a centre, draw the concentric dotted circles round to g, h, i, k, I ; then 
draw parallel lines to the rake, to cut the centre C at 1, 2, 3, 4, 5, and a, h, c, d, e, on the other 
side ; and from these points let lines be drawn perpendicular through the plan. And on the 
centre g of the rib C, square a line up to 5, the top of the arch C ; and from 5 draw a line 
perpendicular through the plan. Also through the points 1, 2,* 3, 4, 5, at B, draw perpendi- 
cular lines to the plan the other way ; begin at h, and trace through the angles both ways, will 
give the place of the angles upon the plan. 

The moulds for bending oveT the angles are found in the same manner as in the last plate, 
by taking the stretch-out round A, and laying it between I) and E. 

The reader may see such groins executed under the Adelphi buildings in the Strand, Lon- 
don, where the declivity is very rapid in going to the river. 

The jack ribs of the groin are cut in the same manner as directed in the last plate, and in 
the practice there will be no occasion for tracing the angles, as the two moulds D and E are 
done independent of them ; the reader will farther observe, that the arch B must not be used 
instead of the arch Af which would produce a very great error in the moulds D and E, as it 
must be evident to every one, that the section upon the square of the cylinder, or body range, 
must be less in the height than the perpendicular or plumb section B, which in this case is ob- 
lique : if these things are properly understood, there will occur nothing in brick groins but 
what may be easily surmounted. 

■ -J* 

In all kinds of brick groins the centres or body ribs must be fixed first in the same manner 

as if there were no side arches cutting across them ; then the centres must be boarded over ; 

then to find the place of the angles upon the boards, that is, the proper intersection of the side 

arches upon the plan, the moulds D and E must be both bent round the boards at one time, by 

keeping the point*; I and e of the moulds D and E upon the tops of the piers at o and e ; then 



40 CRADLING TO GROINS. 

keep the top points together, and bend them round, keeping them still together, then the point 
at 5, will fall perpendicularly over h in the plan ; round the inner edges of the moulds draw a 
curve upon the boards, which will be the proper intersection of the side arch. The jack ribs 
are cut in the same manner as directed in the last. 



PLATE XX. 

The Angles or Diagonals of any Plaster Groin, which are straight upon the Flan, and one 
of the Side Arches, being given, to find the other Side Arch and Angle JRib. 

Fig. 1. Case 1. If the given rib is a semicircle or semi- ellipsis, they may be described 
as in fig. 2, plate 6, with a trammel, which is by far the readiest method; but if a proper 
trammel is not to be got, a temporary one may easily be made, which will answer equally well 
by fixing two pieces of wood in the form of a square, that is, to make a right angle ; each leg 
must be as long as the difference bejtween the semi- transverse and semi-conjugate axis, and 
instead of the sliding nuts in the rod, two brad awls will answer the purpose, being put through 
any straight slip of wood; and by moving this round either the exterior or the interior angles 
of the squar§, keeping the pins or brad awls close to each leg, it will describe one quarter of 
an ellipsis at one time. 

To find the Length of the Jack Bibs. * 

Lay down the plan of the ribs, as at B, and draw a rib upon each opening ; then draw 
perpendicular lines from the plan of each opening, at the extremities ace, to cut its cor- 
responding ribs at b df: then the distance from b to b shows the length of the first jack 
rib, from dtod the length of the second, and from/ to/ the third. 

How to bevel the Angle Bibs, so that they shall range with each opening of the Groin. 

First get the ribs out in two halves or thicknesses, as at JE and F, then draw the 
plan of your angle rib which is placed between E and F, will show the true ranging upon the 
bottom of the rib ; then shift your hip mould parallel upon the base of JS and F, will 



rj,>h 



Fi</ . 1. 




Fu/ . 




f 



I 




Fhitt 21. 




CRADLING TO GROINS. 41 

show how much wood there is to be bevelled off; then nail the two halves together, and it 
will be completed. 

aiETftop I. 

Fig. 2. Case 2. When the given rib is a segment of a circle, or any other curve 
whatever, the ribs will be described as in plate 15, fig. E, as are shown at B, E, and F. 

METHOD II. 

When the given arch is a segment of a circle as at A^ take its height h c, and place it from 
6 to c at C and D ; then take the whole diameter of the arch A, that is, twice the radius a c, 
and place it from the crown of the other arches perpendicular to their bases from c to 6 at C, 
and from c to f? at Z) ; then the arch may be drawn as in plate 8, by intersecting lines : the 
ranging of the ribs is done in the same manner as in the last groin. 

Either of these two methods is much readier in practice than tracing the ribs through 
ordinates. 



PLATE XXL 

Given one of the Body Bibs, and the Angles straight upon the Plan, and the Ascent of a 
Groin not standing upon level Ground, to find the Form of the ascending Arches, and the 
Angle Bibs. 

Let 6 a c at ^ be the angle of the ascent, from the point b make 6 e perpendicular to a h, 
and describe the rampant curve B, as in plate 15, at No. 3, mfig. E ; then draw the diagonal 
a 6 at ^, and make b c perpendicular to it, and equal to 6 c at 5 ; then draw the hypothenuse 
'^%4i c, and describe the angle rib E, in the same manner as that of B. 

To find the Length of the Jack Bibs, so that they shall fit to the Bake of the Groin. 

Draw lines up from the plan to the arch, as at D, in the same manner as explained in the 
last plate ; then the arch from a to a is the first jack rib, from b io b the second, and from 
c to c the third, kc. 

[6] 



42 CRADLING TO GROINS. 



How to range the Jingle Ribs for such sort of Groins. 

Get the ribs out in two halves, as in the last plate, then the bottom of the ribs must be 
bevelled agreeably to the ascent of the groin, and the plan of it must be drawn upon the level, 
and from thence they may be drawn perpendicular from the plan to the rake of the rib : then take 
a mould to the form of the rib, or the rib itself, and slide this agreeably to the rake to the 
distance that is marked upon th€ bottom to be backed off, will show how much the rib is to 
bevel all round. 



PLATE XXII. 

OF GROmS CUTTIJVGUJ^DER PITCH. 

DEFINITION. 

When the side arches of a groin are lower than the body arch, then they are called under 

pitch groins. 



Given one of the Body Ribs B, and the Height i^, of a Door or Window, ^c. at D, and its 
Width m 1, to find the Side and Angle Ribs D and E, so that the Intersection of the Side 
Arch D, with the Body Rib B, shall be straight upon the Plan. 

Draw c e perpendicular to c b, the base of B, and equ^ to the height of the window at 
B, that is, equal to/jg- ; through e draw e a parallel to c b, cutting the arch B ma; let fall the 
perpendicular ab to c b, and continue it so as to cut the Yvae,fg produced to k, and draw k m^f 
and k I, which is the place of the angles upon the plan, or the base of the Sangle ribs ; then the 9 
ribs -D and E may be described from the given rib F, as directed in plate 15, fig. E, from a 
centre ; or they may be described as dXfig. F, of the same plate, as you see on the other side at 
A and C by ordinates ; but the first is by far the easiest method for practice, for if you stick 
a pin or brad awl in g^ at J), and lay a chalk line to it, you may strike all the radical lines 



Plate 2 i>. 




■^ 



Fhih '23. 




CRADLING TO GROINS. 43 

o. 1 o- 2, g-S, g 4, &c. in much less time than the parallel lines in A and C can be drawn, and 
with much greater accuracy ; and the divisions upon c w of the arch F, may be marked upon a 
rod, and readily transferred to the arches Z) and E, on mp, and/^ : then move your brad awl 
out of g-, and stick it in the crown at/, and strike lines from the divisions of mp to cross the 
other lines, will give the points through which the arch must pass ; but the reader must recol- 
lect that four or five points will not be suflS.cient in the practice for tracing the curve with 
accuracy, and therefore a greater number must be found. At the other end of the groin is 
shown the manner in which it may be fixed, sufficiently intelligible for a workman. 



PLATE XXIII. 



A CYLIJYDRO-CYLIJVDRIC ARCH. 



DEFINITION. 



^ cylindro-eylindric arch or Welsh groin is an under pitch groin, whose side and body arches are 
both given semicircles, or they may he similar segments of circles cutting through one ano- 
ther, whose intersections do not meet in a plane surface, that is, the place of the ribs will 
not be sti'aight upon the plan, but will generate a curve line. 



Given the Body Rib A, and the Side Rib B, of a Cylindro- Cylindric Arch, to find a Mould for 

the intersecting Ribs. 

Di^'ide half the arch B into any number of equal parts, 1, 2, 3, d, or they may be taken 
at discretion, and from these points let fall perpendiculars to a b, its base ; produce them at 
pleasure: also from the same points 1, 2, 3, d, draw lines parallel to ab, the base of B, to 
intersect the perpendicular line ef\ transfer the divisions from efio eg; then from the divi- 
sion oi e g draw lines parallel to p q, to intersect the body rib A at the points h u w y: from 
these points draw perpendiculars to p q, its base, and continue them to intersect with the per- 
pendiculars from B, at the points k, I, m, n, between C and D ; then trace a curve through 

6* 



44 CRADLIKG TO GROINS. 

these points, which will be the place of the intersecting ribs upon the plan : then draw two 
other curve lines on each side of klmn^ &c. to make the thickness of the rib upon the plan ; 
on the inside of the curve draw two chords for each half to their extremities, draw two other 
lines parallel to them to touch the outside curve, then the distance between those two straight 
lines will show what thickness of stuff it will take to make the intersecting rib ; through the 
points k Imn, &c. draw perpendicular lines to the chords, make the heights c <f, 3 3, 22, 1 1, 
&c. at D, equal to the corresponding heights at B : then D is the mould for the intersecting 
rib j C is the same as Z). 



, To range the Ribs, so that they will stand perpendicular over the Plan. 

At the points x, v, t, i, in the base of C, draw the parallel dotted lines to the ordinates 
of C and D, and make their corresponding heights equal to those of the arch B ov A', draw 
the dotted curve line huwy zt C, and it will show how much is to be bevelled oif on that 
side of the rib ; in like manner the other side D is bevelled, as is shown by the dotted 
curve line. 



To find a Mould to bend under the intersecting Bibs, so that it shall give the Place of the 

Angle truly upon the Plan. 

Take the stretch round the under side of the rib D at the dots, by bending a thin slip of 
wood round it, mark it at each dot, and stretch it out along the straight line be sX E, draw 
the ordinates across, and prick them from the plan that lies between D and C, then E agree- 
ably to the letters will be the mould required. 



Note. The straight edge of the mould must be kept exactly to the face of the rib ; when it is bent round, 
then draw a curve round the under side of the rib, by the other edge of the mould, will give the true place of the 
angle. . ' 



'•. 



Fhih' 24. 




Phife26. 
Ficj .1. 




F,\, .:?. 




I tii i i ^'T'-'-i^ i ' "in t ffl MB 



MHIP 



CRADLING TO GROINS. 45 



PLATE XXIV. 



There will be no occasion for explaining the lines of this groin, as they are of the same 
nature as those in the last plate ; but it will be proper to take notice, that this is a bevel groin, 
the ribs must lie in the same direction as the plan of the groin, which will make them longer 
than their corresponding given arches at the top, but of the siame height ; they are conse- 
quently ellipses, being the sections of cylinders, therefore to make a rib over Im, across the 
two piers, take the extent of the base Im, and the height of the given arch n o, and describe an 
ellipsis ; and to describe the side arches between any two piers, as from a to 6, take the extent 
a b, and the height of the given arch, p gf at c/5, and describe an ellipsis, it will give the proper 
form of the rib to stand over a b ; the intersecting ribs will require two moulds C and Dy ow- 
ing to the groins being bevel upon the plan. 

J^ote. The letters are marked the same upon D and C as they are upon E, to show they 
are traced from it. 



PLATE XXV. 

To describe the intei'secting or Angle Ribs of a Groin standing upon an octagon Plan, the 
Side and Body Ribs being given both to the same Height. 

Fig. 1. EhsQ. given body rib, which may be either a semicircle or a semi- ellipsis, and 
.^ is a side rib given of the same height ; Z) is a rib across the angles : trace from E, the basis 
of both being divided into a like number of equal parts, divide the base of the given rib vi, 
into the same number of parts ; from these points draw lines across the groin to its centre at 
m, and from the divisions of the base of the other rib JD, draw lines parallel to the side of the 
groin, then trace the angle curves through the quadrilaterals, will be the place of the inter- 
secting ribs ; draw the chords a b and b c, then prick the moulds B and C from E or B, but 
take care not to prick them from the crooked line at the base, but from the straight chords a b 
and be. 



46 CRADLING TO GROINS. 



To describe and range the Jingle Ribs of a Groin circular upon the Plan^ the Side and Body 

Arches being given, as in the last Groin. 

The ribs are described in the same manner as in the last example for the octagon groin, or 
in the same manner as the cylindro-cylindric, plate 23, and the ranging is found in the same 
manner as is described in that plate. 

JVbte. E and F are the same moulds as are shown at B and D. 



PLATE XXVI. 

The Side Rib A, and the Angles being given straight upon the Plan, to find the Angle Rib G, 

' and the Body Rib C. 

Let the rib A be supposed to be placed over the straight line a b, as its base, which divide 
into any number of equal parts, as eight, from the points of division draw lines to the centre of 
the groin to intersect the angles at a, 6, c, d, e,f, g, these points will give the perpendiculars of 
the ordinates of G, which being made respectively equal to those of A, will give the curve of 
the rib G. If from the points a, b, c, &c. arcs be drawn from the centre of the groin to inter- 
sect the base of C, at 1, 2, 3, 4, 3, 2, 1, and perpendiculars be drawn and made correspondingly 
equal to those of .^, and C be traced through these points, then C will be the body rib. 

Hoiv to describe the Ribs of a Groin over Stairs upon a circular Plan, the body Rib being 

given. 

Fig. 2. Take the tread of as many steps as you please, suppose nine, from E, and the 
heights corresponding to them, which lay down a.tF', draw the plan of the angles as in the 
other groins, and take the stretch round the middle of the steps at E, and lay it from a to 6 at 
F; make de perpendicular to dc B.t B, equal to (/e at F, draw the hypothenuse e c, draw per- 
pendiculars from c?c up to B, and prick 5 from A, as the figures direct, then B is the mould to 
stand over a b ; draw the chords a 4 and 4 m at the angles, make a 9, 4 h, perpendicular to 



rh</^' \'7. 




jy^///' r/^c />r/ik/iY/>s 



CRADLING TO NICHES. 47 

them, each equal to half the height de, at B or F, draw the hypothenuse g 4, and hm, draw 
the perpendicular ordinates from the chords through the intersection of the other lines that 
meet at the angles, then trace the moulds D and C, from the given rib .^, will form the moulds 
for the angle or intersecting ribs. 

JVote. The reason that the angle ribs D and C are laid contrary ways, is only to avoid 
confusion. 

PLATE XXVII. 

,.is all the sections of a sphere are circles, and those passing through its centre are equal, and 
the greatest which can be formed by cutting the sphere : it is therefore evident that if the 
head of a niche is intended to form a spherical surface, the most eligible method is to make 
the plane of the back ribs pass through the centre; this may be done in an infinite variety of 
positions; but perhaps the best, and that which would be easiest understood, is to dispose 
them in vertical planes. If the head is a quarter of a sphere, the front rib, and the still 
plate or springing, on which the back ribs stand will curve equally with the vertical ones ; 
but if otherwise they will be portions of less circles. But it is evident if the front and 
springing j'ibs are intended to be arcs less than those of semicircles, either equal to each 
other or unequal, that as they are posited at right angles to each other there can be only one 
sphere which can pass through them; consequently if the places of the vertical ribs are 
marked on the plan these ribs can have only one curve: in the former case no diagram is 
necessary, but in the latter it may be proper to show how the vertical ribs and their situation 
on the front rib g,re found. 

To get out the Ribs for the Head. 

From the centre C draw the ground-plan of the ribs as at figure A, and set out as 
many ribs upon the plan as you intend to have in the head of the nichc;, and draw them all 
out towards the centre at C. Place the foot of your compass in the centre C, and from 
the ends of each rib, at e and c, draw the small concentric dotted circles round to the 
centre rib at m and n ; and draw mg and n i, parallel to rk, the face of the wall ', then from 
q round to c upon the plan, is the length and sweep of the centre rib, to stand over a b ; 
and from i round to e, the length and sweep of the rib that stands from c to d upon 



48 CRADLING TO NICHES. 

the plan ; and from g round to e is the sweep of the shortest rib, that stands from e to/ upon 
the plan. 



Secondly. To bevel the Ends of the Back Ribs against the Front Rib. 

The back ribs are laid down distinct by themselves at C, B and E, from the plan. Take 
c 1, m figure Jl, and set it from c to 1 in B, will give the bevel of the top of the rib B. And 
from figure ./?, take from e to 2 upon the plan, and set from e to 2 in the rib E, will give the 
bevel of the top. 



Thirdly. To find the Places of the Back Ribs where they are fixed upon the Front. 

From the points a, c, and e, at the ends of the ribs, in the plan, figure t/2, draw the dotted 
lines up to the front rib, to df and w, which will show where they are to be fixed upon the front 
rib. The double circle upon the front rib shows the ranging. 



PLATE XXVIII. 

To find the CWve of the Ribs of a Spherical JViche, the Plan and Elevation being give^i 

Segments of Circles. 

In fig. A is the elevation of the niche, being the segment of a circle whose centre is t ; 
at B is the plan of the same width, and may be made to any depth, according to the place it 
is intended for, and its centre is c ; on the plan B, lay out as many ribs as it will require, 
draw them all tending to the centre at c, they will cut the plan of the front rib in g, f, 
e, d ; through the centre c, draw the line m n, parallel to a h, the plan of the front rib ; put 
the foot of your compass in the centre at c, draw the circular lines from a, g, /, e, d, to the 
line m n, and make cs eqnal to u t, that is, make the distance from the middle of the chord 
line mnto s, the centre of the arch at C, equal to the distance from the middle of the chord 
at the top at fig. A, to its centre, at t ; then place the foot of your compass in s, as a centre, 
and from the extremities m or n, describe the arcli at C ; with the same centre draw an- 



ri^t. ?//. 



F,o ..I. 




i - / 



7/;' 




:: ^■•-''-■'' • ' '-l^' 




*i/ ■ 


1 ■ • ^y 





F ,. 




CRADLING TO NICHES. 49 

other line parallel to it, to any breadth as you intend your ribs shall be ; then C is the true 
sweep of all the back ribs in the niche. 

J^ote. The points /, k, i, h, show what length of each rib will be sufficient from the point 
m ; from A to m is the rib that will stand over dx, from i to m is the rib that will stand over 
ew, from A to m overfv^ and from / to m over g w : the other half is the same. 

Through the centre t, draw d E; parallel to a b, complete the semicircle e f g d, then d e is 
the diameter ; through n draw n a parallel to u d, in the centre t, with the distance t a de- 
scribe another semicircle, whose diameter is c b ; then will the semicircle c e g a b, be equal 
to a vertical section of the globe, standing on k i, passing through its centre at c, which is the 
same curve as the rib at C, because u A is equal to c n, and cs bisecting mn a.t right angles, is 
equal to t u, bisecting e a at right angles : therefore the hypothenuse t a, that is, the radius of 
the circle b a g e c, is equal to s w or s n, the radius of the circle or rib at C. 



PLATE XXIX. 

V 
The Plan of a JViche in a ci7'cular Wall being given, to find the Front Rib. 

B is the plan given, which is a semicircle whose diameter is a b, and a, i, k, I, m, h, the 
front of the circular wall ; suppose the semicircle B, to be turned round its diameter a b, so that 
the point v may stand perpendicular over h in the front of the wall, the seat of the semicircle 
standing in this position upon the plan will be an ellipsis ; therefore divide half the arch of B 
upon the plan into any number of equal parts, as five ; draw the perpendiculars ld,2e, 3/, 
^g, 5h, upon the centre c with the radius c h, describe the quadrant of a smaller circle, which 
divide into the same number of equal parts as are round B ; through the points 1, 2, 3, 4, 5, 
draw parallel lines to a b, to intersect the others or the points d, e,/, g, h, through these points 
draw a curve, it will be an ellipsis ; then take the stretch-out of the rib B, round 1, 2, 3, 4, 5, 
and lay the divisions from the centre both ways at F, stretched out ; take the same distances 
di, ek,fl, gm, from the plan, and at jPmake di, ek, fl, equal to them, which will give a 
mould to bend under the front rib, so that the edge of the front rib will be perpendicular to 
a, I, kf /, m. 



50 CRADLING TO NICHES. 

J^ote. The curve of the front rib is a semicircle, the same as the ground-plan, and the 
hack ribs at C i) and E are likewise of the same sweep. 

The reason of this is easily conceived, the niche being part of a globe, the curvature must 
be every where the same, and consequently the ribs must fit upon that curvature 

JVote. The curve of the mould jP will not be exactly true, as the distances di^ ek,fl, &c. 
are rather too short for the same corresponding distances upon the soffit at jP, but in practice it 
will be sufficiently near for plaster work ; but t^pse who would wish to see a method more ex- 
act, may examine plate 15, Jig. A, where C is the exact soffit that will bend over its plan at B. 

In applying the mould Pwhen bent round the under edge of the front rib, the straight side 
of the mould F must be kept close to the back edge of the front rib, and the rib being drawn 
by the other edge of the mould, will give its place over the plan. 



PLATE XXX. 

The Plan and Elevation of an elliptic JViche being given, to find the Curve of the Ribs. 

Fig. ,d. Describe every rib with a trammel, by taking the extent of each base from the 
plan whereon the ribs stand to its centre, and the height of each rib to the height of the top of 
the niche, it will give the true sweep of each rib. 

To back the Ribs of the JViche. 

There will be no occasion for making any moulds for these ribs, but make the ribs them- 
selves ; then there will be two ribs of each kind ; take the small distances 1 e, 2 c?, from the 
plan at B, and put it to the bottom of the ribs D and E, from d to 2, and e to 1 ; then the 
ranging may be drawn off by the other corresponding rib ; or with the trammel, as for example, 
at the rib E, by moving the centre of the trammel towards e, upon the line e c, from the centre 
c, equal to the distance 1 e, the trammel rod remaining the same as when the inside of the curve 
was struck. 



P/o^'JO 



Fin . A 




w 



50 CRADLING TO NICHES. 

J^ote. The curve of the front rib is a semicircle, the same as the ground-plan, and the 
back ribs at C i) and E are likewise of the same sweep. 

The reason of this is easily conceived, the niche being part of a globe, the curvature must 
be every where the same, and consequently the ribs must fit upon that curvature. 

. v^ • 

JVb^e. The curve of the mould jP will not be exactly true, as the distances di, e k, fl^ &c. 
are rather too short for the same corresponding distances upon the soffit at F, but in practice it 
will be sufficiently near for plaster work ; but t^ose who would wish to see a method more ex- 
act, may examine plate 15, fig. A, where C is the exact soffit that will bend over its plan at B. 

In applying the mould i^ when bent round the under edge of the front rib, the straight side 
of the mould F must be kept close to the back edge of the front rib, and the rib being drawn 
by the other edge of the mould, will give its place over the plan. 



PLATE XXX. - . 

The Plan and Elevation of an elliptic JViche being given, to find the Curve of the Ribs. 

Fig. a. Describe every rib with a trammel, by taking the extent of each base from the 
plan whereon the ribs stand to its centre, and the height of each rib to the height of the top of 
the niche, it will give the true sweep of each rib. 

To back the Ribs of the JViche. 

There will be no occasion for making any moulds for these ribs, but make the ribs them- 
selves ; then there will be two ribs of each kind ; take the small distances 1 e, 2 c?, from the 
plan at B, and put it to the bottom of the ribs D and E, from d to 2, and e to 1 ; then the 
ranging may be drawn off by the other corresponding rib ; or with the trammel, as for example, 
at the rib E, by moving the centre of the trammel towards e, upon the line e c, from the centre 
c, equal to the distance 1 e, the trammel rod remaining the same as when the inside of the curve 
was struck. 



Fin. A 



Plate 30 




Flak 31 




A I; 




Fui.Q. 



f! .•-•-.- 





CRADLING TO DOMES, &c. 51 

Given one of the common Bibs of the bracketing of a Cove^ to find the ^ngle Bracket for a 

rectangular Room. Fig. F. 

Let ^be the common bracket, 6c its base; draw ba perpendicular to be, and equal to 
it draw the hypothenuse a c, which will be the place of the mitre ; take any number of ordi- 
nates in H, perpendicular to b c, its base, and continue them to meet the mitre line a c, that is, 
the base of the bracket at /; draw the ordinates of / at right angles to its base ; then the 
bracket at /, being pricked from H, as may be seen by the figures, will be the form of the 
angle rib required. 

J^ote. The angle rib must be ranged either externally or internally, according to the 
angle of the room. 

Having given a common Bracket K, Fig. ^,for Plaster Cornice, to find the Mitre Bracket L. 

Proceed as in the last example, and you will have the bracket required. 



PLATE XXXI. 

0¥ PEJ^DEJYTIVES AJVD INTERIOR DOMES WHEN PLACED OVER THE OPEmJVGS 

OF ROOMS. 



One of the Bibs of a Dome being given, and the Plan of the Opening of a Staircase which is 
square, and aii octagon Curb at the Top for a Sky-light ; to find the Bibs and the springs 
ing Curve on each Side of the opening of the Staircase, where the Foot of the Bibs come, so 
that Part of the Lome shall be an octagon finish, agreeably to the Curb. 

Fig. a. Let B be the given rib ; take any number of perpendicular ordinates to its 
base at pleasure, from the points a, c, e, g, i, I, where they intersect its base, draw parallel 
lines to the sides of the curb, returning round each diagonal, if there is more than one, till 
it cut the base of the angle rib D ; at the points a, c, e, g, i, I, draw the ordinates of D, and 

7* 



52 CRADLING TO DOMES, &c. 

prick it from B, will be the angle rib ; and at the points e, g, i^ /, at C, upon the side of the 
opening of the staircase, draw the perpendicular ordinates, and prick C from B, agreeably to 
the letters ; then the curve C will be the true place for the foot of the ribs upon the side of the 
staircase, and the part that lies in the middle is a straight line parallel to the horizon. 



The Opening or Plan of the Room being a Square as before, and the vertical Section of a 
semicircular^ to find the springing Curve D on the Side of the Boom for the Foot of the 
Bibs, so that it shall finish to a circular Curb at the Top. 

On the side of the staircase Im, as a diameter, describe a semicircle ; JD will be the true 
place for the foot of the ribs ; this is evident, for every section of a semi-globe, at right angles 
to its base, is a semicircle, and this is the same thing if truly considered. 

JVbte. All the ribs of this dome are cut by the rib at C, as explained by the perpendicular 
lines ; draw round the centre a, from the points of each bracket, at c (? ef to the points kihg, 
from these points draw perpendicular dotted lines, and these will show what length each bracket 
must have according to its place. 



The vertical Section of a Segment Dome passing through its Centre being given, the Plan of 
the Opening of the Boom being still a Square, as before, to find the Section upon each 
Wall for the Springing of the Bibs, to finish to a circular Curb at the Top. 

Let the section D across the angle be given, whose centre is k, and the distance of the 
centre from the chord k I ; bisect the side c g oi the wall b h, at right angles at the point i ; 
from i make i b equal to /A ; with a radius b g or be, describe the segment c, m g will be the 
true place of springing of the ribs ; all the other lesser ribs are cut from the angle rib D : all 
this is evident from the sections of a globe, which is already described in the Geometry. 

Fig. D is of the same nature as the others, having an ogee top ; the section F is traced 
from E. 



Flate32. 






CRADLING TO DOMES. 55 

PLATE XXXII. 

Fig. r/2 is the plan of an feUiptical domical sky-light over a staircase; 5 and C are the 
sections, which show how to place your ribs. 

How to proportionate the Length of the inside Curb to- any Width given. 

Proceed as directed in page 58 for an elliptical dome, that will determine the true 
length to the width. 

How to proportionate the circumscribing Ellipsis, to pass through the Angles at a, b, c, and 
d, to have the same Proportion as a b, and he, of the Sides of the Staircase, 

Proceed as directed in fig, 5, plate 6, in the Geometry. 

To describe the Ribs, . 

The rib over from 7i, to the centre of the trammel in fig. A, is a given quarter of a circle, 
as is shown at F, and of course all the other ribs must come to the same height with it. 
Suppose it was required to find a rib over dp, you must take the full extent from d to the 
centre, and describe the quarter of an ellipsis D ; then the part over dp will be as much 
of it as is wanted : in the same manner E will be described, and the part over i o is what 
is wanted of this rib ; the same letters are marked upon the bases of D and E, as they are 
in the plan, fig. A. Every other rib is described in the same manner. 

To find the Springings on each. Side of the Room for the Foot of the Ribs to stand upon. 

Describe the semicircle C, to c b, the width of the room, and it will give the bottom of the 
ribs on that side ; and describe a quarter of the ellipsis B, for the bottom of the ribs on the 
other side, to the same height as C. 

This method depends on this principle, that all the parallel sections of a spheroid are 
similar figures : therefore a vertical section standing upon a b, will be similar to a vertical 
section passing through its centre ; both will be similar ellipses ; but a 6 is an ordinate to 



54 ROOFING. 

the conjugate axis, and 6 c is an ordinate to the transverse of the circumscribing ellipsis ; by- 
construction half the length of the parallelogram is to half the length of the ellipsis, as half 
the width of the parallelogram is to half the width of the ellipsis, and a spheroid may be 
supposed to be generated by the revolution of a semi-ellipsis about its axis ; hence it follows, 
that all sections of a spheroid parallel to the axis are similar figures, consequently the section 
B is similar to the circumscribing ellipsis of the ground plan. 



PLATE XXXIII. 

Let a 6 be the end of ^ a' rectangular roof, ae and bf being a part of each side, let ae 
and bf be each equal to half the width a 6 of the roof; join e/; bisect ef in c ; draw cd 
perpendicular to ef; *and make cc? equal to the height of the roof; joinc?e and df and de 
and df are the length of the principal rafters ; join ac and be ; produce either diagonal, as 
b eto g, make c g equal to c c^; join ag, and a^ is the length of each hip. 

Draw any line a? y perpendicular to the seat a c of a hip, cutting aeand ab at x andy, 
and a c at z: from z describe a tanged circle to a g, cutting a e at ly ; join w x and w y, 
and the angle i/ w x is the inclination of the planes which form the hip angle, and is what is 
generally termed the backing of the hip. 

In this plate one end of thereof is shown in order to show two cases: the first is when 
the purline lies level, or having two sides parallel to the horizon ; the square at B, and the 
bevel at C, will show how to draw the end of the purline in this easy case; but the following 
method is universal in all positions of the purline. 

JVbfe. There will be no occasion to draw this at large ; as the bevels will be the same if 
done to ever so small a scale, and the sides may be measured from the scale. 

To find the Bevels of a Purline against the Hip Rafter. 

Let the purline be in any place of the rafter, as at I, and in its most common position, 
that is, to stand square, or at right angles to the rafter ; and from the point h as a centre, 
with any radius describe a circle. Draw two lines q I, and p n, to touch the circle in^ and q, 
parallel to/6 ; and at the points s and r, where the circle and two sides of the purline inter- 



Flace 33. 




Width of the Jtoof 



Tje Beam 

\o in \v •> in 1 




nvii rhite 





FLi/. i / 




r/a/v JA 




ROOFING. % SS 

sect, draw two parallel lines to the former, to cut the diagonal in m and k ; and draw m n 
and k I perpendicular to s m and r k, and join the points n, i and A, i ; then G is the down 
bevel, and F the side bevel of a purline : these two bevels, when applied to the end of the 
purline, and when cut by them, will exactly fit the side hip rafter. 

To find the Bevels of a Jack Rafter against the Hip. 

By turning the stock of the side bevel of the purline, at F, from a round to the line i z, 
will give the side bevel of the jack rafter. And the bevel at .^, that is, the top of a com- 
mon rafter, is the down bevel of the jack rafter. 

At the bottom is shown the manner of cocking down the tie beam upon the wall plate ; 
the proper size of the cocking is figured. 



PLATE XXXIV. 

This plate shows the manner of framing a roof in ledgement ; but as roofs are seldom exe- 
cuted in this manner, I shall not be very particular in describing its lines. The following 
description for winding will serve for any. 



PLATE XXXV. 

Hffw to lay out an irregular Roof in Ledgement, with all its Beams lying bevel upon the Plan, 
so that the Ridge may be level when finished ; the Plan and Height of the Room being given. 

The lengths of the common and hip rafters are found as usual. From each side in the 
broadest end of the ropf, through c and d, draw two parallel lines to the ridge line ; draw 
lines from the centres and ends of the beams perpendicular to the ridge line, and lay out 
the two sides of the oof JD and E, by making e d &t E equal to a? w in j3, the length of the 
longest commoB rafter, and cain^ equal to mw at .^, and so on with all the other rafters. 



56 ROOFING. 



To find the Winding of this Roof . 

Take y v half the base of the shortest rafter : and apply this to the base of the longest 
rafter from z to 1 ', then the distance from 1 to 2 shows the quantity of winding. 



How to lay the Sides in Winding. 

. Lay a straight beam along the top ends of the rafters at JS, that is, from c to e, and lay 
another beam along the line a b, parallel to it, to take the ends of the hip rafters of m and 
If and the beams to be made out of winding at first. Raise the beam that lies from a to b, 
at the point b, to the distance 1 2 above the level ; which beam, being thus raised, will raise 
all the ends of the rafters gradually, the same as they would be when in their places. 

The same is to be understood of the other side D ; the ends are laid down jn the same 
manner as making a triangle of any three dimensions. 

To satisfy the curious, I have given the lines of this roof 5 but in practice there is not the 
least occasion for framing the sides in winding ; for, instead of the ridge line, the top is 
made level at the widest end of the roof, from the narrowest end, which begins at a point ; 
and by this means the sides may be framed quite out of winding, which will have a much 
better effect than any winding roof can have. 



PLATE XXXVI. 



POLYGOJV ROOFS. 



The methods of constructing regular polygons upon any given side, are shown at^^. 
4, 5, and 6, in Geometry. 



/y^///- JO 




ROOFING. 57 

The Phn of a polygon Roof being given, and one of the common Bibs standing upon that 
Plan, to find the Angle Rib, and the Form of the Boards that will cover it when the Ribs are 
fitted up. 

In fig. A let B be the given rib ; divide the curve in any number of equal parts, as 
four, and lay them at B from a to 4, which bisects b b, the side of the polygon, at right 
angles ; through these points draw lines parallel to the side bb Q,i the polygon ; at B and 
Z) make 1 c at Z) equal to c c, between B and C make 2 d equal to d d, and 3 e equal to e e, 
&c. and through the points b, c, d, e, f, draw a curve line, which will be the form of the 
boarding ; from the points g, f, e, d, c, draw lines at right angles to g b, the base of the angle 
rib, and prick the rib C from B as they are marked by the letters, which is plain. 

J^'ote. The more parts there are in this operation, the truer will it be, or any other of 
this nature. 

In the same manner may the covering and angle ribs of any other polygon be found, 
whatever may be the form of the ribs, as is shown at figures B and C. 

To find the Covering of a spherical Dome. 

Fig. jD. Make a circle i ckf, of the plan of the dome, and if it is a semi-globe take 
the stretch-out of one quarter for the length of a board j make the length of K from a to 
4 equal to it, and let c c, at the bottom, be any breadth that the board will admit of ; on the 
base c c, as a diameter, make a semicircle ; divide half the arch line into any number of equal 
parts ; draw the little lines 11, 2 2, 3 3, parallel to c c, the base of the board, and divide 
the height into the same number of equal parts ; draw the ordinates across 5 make 11,2 2, 
3 3, upon these ordinates, equal to 1 1, 2 2, 3 3, in the semicircle at the bottom ; a curve 
being drawn through these points will be the mould ^for the covering. 



[8] 



58 ROOFING. 



To cover a spherical Dome when the Top does not rise so high as a Semicircle^ but only a 

Segment. 

Suppose Id to be the height of the dome at F^ and the width cf of the dome as before, 
upon the chord c/, with the perpendicular height / c? describe a segment, which will be the 
same as a vertical section standing upon cf; here is only one half of the segment, which is suf- 
ficient: draw the chord c d; take c« equal to half the width of a board, whatever it will ad- 
mit of : draw a b perpendicular to cut the chord c d dXb ; take the stretch or circumference 
of the arch c d, and make the length of /from a to 4 equal to it ; take the double of a c, at F, 
and make it the base of the board at /; take a b from F, and set it upon the base of I, upon 
the middle of c c, from atob', and with the chord c c, and the height a b, describe a seg- 
ment upon the bottom of the board at /; divide one half into any number of equal parts ; 
likewise divide the height of the board /into the same number of equal parts ; draw ordinates 
in both, and the board / will be completed, as in the same manner is that of H, described 
before. 



PLATE XXXVII. 

CIRCULAR DOMES. 

As the common method of finding the centres for describing the boards to cover a 
horizontal dome will be found in practice very inconvenient, for those boards which come 
near to the bottom ; I shall in this place show how to remedy that inconvenience. 

To find the Sweep of the Boards on the Top. Fig. A. 

Divide round the circumference of the dome into equal parts at 1, 2, 3, 4, 5, 6, &c. each 
division to the width of a board, making proper allowance for the camber of each board ; 
draw a line through the points 1, 2, to meet the axis of the dome at a?; on x, as a centre, 
with the radii a; 1 and ic 2 describe the two concentric circles, it will form the board G ; in 
the same manner continue a line through the points 2 and 3 at C, to meet the axis in w ', 



^/ate :>7. 



V:i /'Vyy A 




^^.^ 




ROOFING. 59 

then to is the centre for the board C ; proceed in the same manner for the boards D, E, 
and F. 

Now suppose F to be the last board that you can conveniently find a centre, for want of 
room ; on Hts centre, and the radius t 5, make from t on the axis of the dome t «, equal to 
t 5 ; through the points 5 and a draw the dotted line 5 ab, to cut the other side of the cir- 
cumference of the dome at b ; from the points 6, 7, 8, 9, 10, 11, draw radial lines to &, to cut the 
axis of the dome at i, k,l, m,n,o', also through the points 6, 7, 8, 9, 10, 1 1, draw the parallels 
6 c, 7 </, 8e, &c. then will each of these parallel lines be half the length of a chord line for 
each board ; then take c 6 from^^. A, which transfer to No. 1, from c to 6 and 6 ; make the 
height c i, at No. 1, equal to c i, 3it Jig. A ; and draw the chords i 6 and i 6 ; then upon either 
point 6, as a centre with any radius, describe an arch of a circle 12; divide it into two equal 
parts at 1, and through the points 6 and 1 draw 6 q ; bisect z 6, va.p ; drawj& q perpendicular ; 
then i 6 is the length, and p q the height of the board G, which may be described as in fig. 4, 
plate 5, of the Geometry. The reader must observe, that the length of a board is of no con- 
sequence so as the true sweep is got, which is all that is required. Proceed in the sam© 
manner with No. 2, by taking d7 from fig. A, and place it at No. 2, on each side of c? at 7 and 
7, and take dk, from fig. j1, and make dka.t No. 2, equal to it ; draw the chords k7 and 
A 7, and bisect k7atn; draw n a perpendicular ; upon the other extremity at 7, as a centre, 
describe an arch 12, and bisect it at 1, and through the points 7 and 1 draw the line 7 a, 
to cut the perpendicular ?i a at a ; but if the distance A 7 is too long for the length of a board, 
bisect the arch 1 at 6 ; through 7 and b draw 7 t, and draw the little chord a 7, and bisect 
it at / ; draw t u perpendicular to intersect 7 4 at m ; and with the chord 7 a and the height t u, 
describe the segment H. 

la the same manner may the next board / be found, and by this means you may bring 
the sweep of your board into the smallest compass, without having any recourse to the 
centre. 
% 

Suppose it were required to draw a Tangent from 8 at Ab. 3, without having recourse to 

the Centre. 

Bisect the arch 8 /8 at / ; on 8 as a centre, with a radius 8 /, describe an arch elt', make 
It equal to /e ; draw the tangent ^8. 

8 * 



60 ROOFING. 



Given three Points in the Circumference of a Circle, to find any JYumber of equidistant Points 
beyond those that will be in the same Circumference. 

Fig. K. Suppose the three points «, 6, c^ to be given ; to one of the extreme points a join 
the other two points b and c by the lines a b and a c ; with a radius a b, and the centre a, de- 
scribe the arch of a circle 612 3; then take b\, and set it from 1 to 3, and from 2 to 3 ; 
through the points 2 and 3, draw a d and a e ; then take b c, put the foot of your compass in 
c, and with the other foot cross the line addXd; with the same extent put the foot of your 
compass in d, and with the other foot cross the line a e at e ; in the same manner you may 
proceed for any number of points whatever. 



PLATE XXXVIII. 



SPHEROIDAL DOMES. 

Fig. j1 is the plan of a spheroidal dome ; B is the longest section, C the shortest sec- 
tion ; at a a in Bj and bb in C, shows how to square the purlines, so that one side may be 
fair with the surface of the dome; the dotted lines from aain B, and b b in C, show how 
to get the length and width of the purline in fig. A ; but if the sides of the purline were made 
to stand perpendicular over the plan, the curve of it would be found in the same manner as 
before ; then it would require no more than half the stuff that the other would, and take only 
half the time in doing, which is a considerable advantage. 

How to proportionate the inside Curb for the Sky-light, so as it shall answer to the Surface 

of the Dome. 

Draw the diagonal i /and km in fig. Jl, and let A c or g/ be the width, then h gov ef 
will be the true length of the curb; because every section parallel to the base will be propor- 
tional to the base. 



/y./A .;/; 





ROOFING. 61 



To find the Ribs for this Dome. 

The ribs in this are got in the same manner as the ribs for a niche, as directed in page 
50 ; and if the reader understands that, he must know this. 

To find the Form of a Board to stand in any Place of the Dome, in order to be bent up to 

the Crown. 

Divide one quarter of the base of the dome at D into three equal parts ab, be, cd^ and 
suppose you would find a board over ab c in the plan ; draw a c, be, c c, and d c, to the centre 
at c ; then take the triangle ab c in D, and lay it down at ab c in G ; then draw the line 
c 1 1 1, &c. at right angles to a b, and describe a rib G to the height of the dome, and the 
length to the perpendicular of the triangle a be, and divide it into five equal parts, lay 
them along the line 111, &c. in H, and prick the mould H from the triangle ab c, as the 
letters are marked. The board K will be found in the same manner. 

JVbte. In the practice, you are to divide one quarter of this dome into as many parts 
as you think the breadth of a board will contain ; and the boards, when got out by this me- 
thod, will fit to a very great exactness ; this is only into three, that the parts may be clearly 
seen to learners. 

If the boards are got out for one quarter to the lines here laid down, the boards that are 
in the other three quarters will not require any other lines, for every board in the first 
quarter will be a mould for three more boards. 



INTRODUCTION 



TO 



PRACTICAL CARPENTRY. 



OF THE COMPARATIVE STRENGTH OF TIMBER. 



PROPOSITION I. 

THE strengths of the different pieces of timber, each of the same length and thickness are 
in proportion to the square of the depth ; but if the thickness and depth are both to be consi- 
dered, then the strength will be in proportion to the square of the depth, multiplied into the 
thickness ; and if all the three dimensions are taken jointly, then the weights that will break 
each will be in proportion to the square of the depth multiplied into the thickness, and divided 
by the length ; this is proved by the doctrine of mechanics. Hence a true rule will appear 
for proportioning the strength of timbers to one another. 

RULE. 

Multiply the square of the depth of each piece of timber into the thickness ; and each product 
being divided by the respective lengths, will give the proportional strength of each. 

EXAMPLE. 

Suppose three pieces of timber, of the following dimensions : 
The first, 6 inches deep, 3 inches thick, and 12 feet long. 



64 



INTRODUCTION TO 



The second, 5 inches deep, 4 inches thick, and 8 feet long. 

The third, 9 inches deep, 8 inches thick, and 15 feet long. The comparative weight that 
will break each piece is required. 





OPERATIONS. 




First. 
6 deep 
6 


Second. 
5 deep 
5 


Third. 
9 deep 
9 


36 
3 thick 


25 
4 thick 


81 
8 thick 


Length 12)108 
Q 


Length 8)100 

12 and a half 


Length 15)648(43 and a fifth 
60 




48 
45 



Therefore the weights that will break each are nearly in proportion to the numbers 9, 12, 
and 43 leaving out the fractions, in which you will observe, that the number 43 is almost 5 
times the number 9 ; therefore the third piece of timber will almost bear 5 times as much 
weight as the first; and the second piece nearly once and a third the weight of the first piece ; 
because the number 12 is once and a third greater than the number 9. 

The timber is supposed to be every where of the same texture, otherwise these calculations 
cannot hold true. 

PROPOSITION II. 

Given the length, breadth, and depth of a piece of timber ; to find the depth of another 
piece whose length and breadth are given, so that it shall bear the same weight as the first 
piece, or any number of times more 



RULE. 



Multiply the square of the depth of the first piece into its breadth, and divide that product 
by its length : multiply the quotient by the number of times as you would have the other 



PRACTICAL CARPENTRY. 65 

piece to carry more weight than the first, and multiply that by the length of the last piece, and 
divide it by its width ; out of this last quotient extract the square root, which is the depth 
required. 



EXAMPLE I. 

Suppose a piece of timber 12 feet long, 6 inches deep, 4 inches thick ; another piece 20 
feet long, 5 inches thick ; requireth its depth, so that it shall bear twice the weight of the first 
piece. 

Proof. 
6 deep 9.7 

6 9.7 



36 67.9 

4 873 



12)144 . . 94.09 

1.91 remainder added 

12 '■ — 

2 times 96.00 

5 

24 



20 length 20)480 



5)480 24 



96)9.7, or 9.8, nearly for the depth 
81 



187)1500 
1309 



191 



EXAMPLE IL 



Suppose a piece of timber 14 feet long, 8 inches deep, 3 inches thick; requireth the 
depth of another piece 18 feet long, 4 inches thick, so that the last piece shall bear five times 
as much weight as the first. 

[9] 



66 INTRODUCTION TO 

S As the length of both pieces of timber is divisible 

by the number 2, therefore half the length of each is 

64 used instead of the whole; the answer will be the 



3 



same. 



half 7) 192 



27.4, &c. 
5 times 

137 

9 half the length 



4)1233 



308.25(17.5 the depth nearly 
1 

27)208( 
189 

345).1925, &c. 



PROPOSITION III. 

Given the length, breadth, and depth of a piece of timber ; to find the breadth of another 
piece whose length and depth is given, so that the last piece shall bear the same weight as the 
first piece, or any number af times more. 



RULE. 

Multiply the square of the depth of the first piece into its thickness ; that divided by its 
length, multiply the quotient by the number of times as you would have the last piece bear 
more than the first ; that being multiplied by the length of the last piece, and divided by the 
square of its depth, this quotient will be the breadth required. 

EXAMPLE I. 

Given a piece of timber 12 feet long, 6 inches deep, 4 inches thick ; and another piece 16 
feet long, 8 inches deep ; requireth the thickness, so that it shall bear twice as much weight 
as the first piece. 



PRACTICAL CARPENTRY. fif 



6 

6 

36 

4 



3)144 

48 
2 

96 

4 

8)384 



8)48 

6 thickness 



Or this at full length, 

6 depth of the first piece 
6 


36 

4 thickness of the first piece 


Length 12)144 


12 
2 by the number of times stronger 


24 

16 length of the last piece 


144 
24 


8)384 '^ 


8)48 


6 thickness 


EXAMPLE II. 



Given a piece of timber 12 feet long, 5 inches deep, 3 inches thick ; and another piece 14 
feet long, 6 inches deep ; requireth the thickness, so that the last piece may bear four times 
as much weight as the first piece. 



5 
5 

25 



12)75 



6.25 
4 



25.0 
14 



100 
25 



6)350 
6)58.333 
9.722 
9 * 



66 INTRODUCTION TO 



PROPOSITION IV. 



If the stress does not lie in the middle of the timber, but nearer to one end than the other, 
the strength in the middle will be to the strength in any other part of the timber, as 1 divided 
by the square of half the length is to 1 divided by the rectangle of the two segments, which 
are parted by the weight. 



EXAMPLE I. 

1 r 

Suppose a piece of timber 20 feet long, the depth and width is immaterial ; suppose the 
stress or weight to lie five feet distant from one of its ends, consequently from the other end 

1 11 1 

15 feet, then the above proportion will be = : = — as the strength at 

10 X 10 100 5 X 15 75 

100 100 1 

five feet from the end is to the strength at the middle, or ten feet, or as = 1 : == 1 -. 

100 75 3 

Hence it appears that a piece of timber 20 feet long is one- third stronger at 5 feet distance 

from the bearing, than it is in the middle, which is 10 feet, when cut in the above proportion. 



EXAMPLE II. 

Suppose a piece of timber 30 feet long; let the weight be applied 4 feet distant from one 
end, or more properly from the place where it takes its bearing, then from the other end it 

111 1 

will be 26 feet, and the middle is 15 feet; then, — = : — = or as 

15 X 15 225 4 X 26 104 

225 225 17 1 

= 1 : = 2 or nearly 2 -. 

225 104 104 6 

Hence it appears that a piece of timber 30 feet long will bear double the weight, and one- 
sixth more, at four feet distance from one end, than it will do in the middle, which is 15 feet 
distant. 



PRACTICAL CARPENTRY. 69 



EXAMPLE IIL 

Allowing that 266 pounds will break a beam 26 inches long, requireth the weight that 
will break the same beam when it lies at 5 inches from either end ; then the distance to the 
other end is 21 inches ; 21 x 5 = 105, the half of 26 inches is 13 .-. 13 x 13=169 ; therefore 
the strength at the middle of the piece is to the strength at 5 inches from the end, as 

169 169 169 

:: or as 1: the proportion is stated thus : 

169 105 105 

lb. 
169 

1 : :: 266 : to the weight required, 

105 

169 



2394 
1596 
266 

105)44954(428 
420 

295 
210 

854 
840 

14 

From this calculation it appears, that rather more than 428 pounds will break the beam 
at 5 inches distance from one of its ends, if 266 pounds will break the same beam in the 
middle. 

By similar propositions the scantlings of any timber may be computed, so that they shall 
sustain any given weight ; for if the weight one piece will sustain be known, with its di- 
mensions, the weight that another piece will sustain, of any given dimensions, may also be 
computed. The reader must observe, that although the foregoing rules are mathematically 
true, yet it is impossible to account for knots, cross-grained wood, &c. such pieces being not 
so strong as those which are straight in the grain j and if care is not taken in choosing the 
timber for a building, so that the grain of the timbers run nearly equal to one another, all 



70 INTRODUCTION TO 

rules which can be laid down will be baffled, and consequently all rules for just proportion will 
be useless in respect to its strength. It will be impossible, however, to estimate the strength 
of timber fit for any building, or to have any true knowledge of its proportions, without some 
rule ; as without a rule every thing must be done by mere conjecture. 

Timber is much weakened by its own weight, except it stand perpendicular, which will be 
shown in the following problems ; if a mortice is to be cut in the side of a piece of timber, it 
will be much less weakened when cut near the top, than it will be if cut at the bottom, pro- 
vided the tenon is drove hard in to fill up the mortice. 

The bending of timber will be nearly in proportion to the weight that is laid on it; no 
beam ought to be trusted for any long time with above one-third or one-fourth part of the 
weight it will absolutely carry : for experiment proves, that a far less weight will break a piece 
of timber when hung to it for any considerable time, than what is sufficient to break it when, 
first applied. 

PROBLEM I. 

Having the length and weight of a beam that can just support a given weight, to find the 
length of another beam of the same scantling that shall just break with its own weight. 
Let I = the length of the first beam, 
L = the length of the second ; 
a = the weight of the first beam, 
w ^= the additional weight that will break it. 
And because the weights that will break beams of the same scantling are reciprocally as their 
lengths, 

11 a 2 

therefore — : — \\w -V-\ / = JF == the weight that will break the greater beam ; be- 

/ jL 2 Z . 

a 
cause ly -f - is the whole weight that will break the lesser beam. 
2 

But the weights of beams of the same scantling are to one another as their lengths : 



PRACTICAL CARPENTRY. 71 

a '. La 

Whence, i : L :: - = ^half the weight of the greater beam. 

2 21 

Now the beam cannot break by its own weight, unless the weight of the beam be equal to 

the weight that will break it : 

a 
w + - 
La 2 2w + a 

Wherefore, = / = I 

2,1 L 2L 

L'a = 2w + axl^ 
.'. a :2w + a :: l^: Z**, consequently V L^ = L = the length of the beam that can just 
sustain its own weight. 

PROBLEM IL 

Having the weight of a beam that can just support a given weight in the middle ; to find the 

depth of another beam similar to the former, so that it shall just support its own weight. 

Let d = the depth of the first beam ; 

X = the depth of the second ; 

a = the weight of the first beam ; 

w = the additional weight that will break the first beam ; 

a 2w + a 
then will w + - ov = the whole weight that will break the lesser beam. 

2 2 

And because the weights that will break similar beams are as the squares of their lengths, 

2w-^a 2 x^ y. w x^ ■\- a 

... 0}\x'v. : = W 

2 2d 

the weights of similar beams are as the cubes of their corresponding sides : 

a ax^ 
Hence d^ '. oi? \'.-\ — - = W 
2 2 c?' 

ax 2ar^w + x^ a 



2d' ■ 2d^ 

ax = 2w + ax d 

.'. a : a + 2w :: d : X = the depth required. 



f2 INTRODUCTION TO 

As the weight of the lesser beam is to the weight of the lesser beam together with the ad- 
ditional weight ; so is the depth of the lesser beam, to the depth of the greater beam. 

Jfote. Any other corresponding sides will answer the same purpose, for they are all pro- 
portioned to one another. 

EXAMPLE. 

Suppose a beam whose weight is one pound, and its length 10 feet, to carry a weight of 

399.5 pounds, requireth the length of a beam similar to the former, of the same matter, so that 

it shall break with its own weight. 

here a = 1 

and w = 399.5 

then a -f 2 w> = 800 = 1 -f 2 X 399.5 

c/ = 10 

Then by the last problem it will be 
1 : 800 :: 10 
10 



I 



8000 = X for the length of a beam that will break by its own weight. 



PROBLEM III. 

The Weight and length of a piece of timber being given, and the additional weight that 

will break it, to find the length of a piece of timber similar to the former, so that this last piece 

of timber shall be the strongest possible : 

Put / = the length of the piece given, 

w = half its weight, 

W = the weight that will break it ; 

X = the length required. 

Then, because the weights that will break similar pieces of timber are in proportion to the 

squares of their lengths, 

Wx^ + wx 
,\P:x^ :: W+ w: = the whole weight that breaks the beam ; 



PRACTICAL CARPENTRY. 73 

and because the weights of similar beams are as the cubes of their lengths, or any other corres- 
ponding sides, 

wx 
then l^ '. x^ WW \ — — the weight of the beam ; 
J? 

IFx^ + 100^ wx^ 
consequently is the weight that breaks the beam = a maximum ; 

therefore its fluxion is nothing. 

3w:j(^x 
that is, 2 Wx x + 2wxx = nothing. 



/ 



3wx 
2 W+2w = 



I 

^W= + ^w 

therefore, x = I x 

3 IV 

Hence it appears from the foregoing problems, that large timber is weakened in a much 
greater proportion than small timber, even in similar pieces, therefore a proper allowance must 
be made for the weight of the pieces, as I shall here show by an example. 

Suppose a beam 12 feet long, and a foot square, whose weight is 3 hundred weight, to be 
capable of supporting 20 hundred weight, what weight will a beam 20 feet long, 15 inches 
deep, and 12 thick, be able to support? 

12 inches square 15 

12 15 

144 75 

12 15 

12)1728 ' 225 

12 

144 



2.0)270.0 

— •»— 
135 



[10] 



74 



INTRODUCTION TO 



But the weights of both beams are as their solid contents ; 



therefore 12 inches square 
12 

144 

144 inches = 12 feet long 


15 deep 
12 wide 

180 

240 length in inches 


576 
576 

144 


7200 
360 




43200 solid contents of the 

144::135::21.5 by prop. 1 
21.5 


20736 solid contents of the 1st beam 
20736:43200::3 
3 

cwt. lb. 

20736)129600(6 .. 28 = the weight of the 
124416 beam 


J 2d 67.5 
135 
270 


5184 


112 


12)2902.5 


10368 
5184 
5184 


12)241.876 


20.15625 
112 


20736)580608(28 
41472 


31250 
15625 
15625 


165888 
I658R8 




17.60000 
16 

30 
5 





8.0 



21 cwt. 56 lb. is the weight that will break the first beam, and 20 cwt. 17 lb. 8 oz. the 
weight that will break the second beam ; deduct out of these half their own weight. 



20::17::8 
3::14::0 half 



ir...3..8 



PRACTICAL CARPENTRY. 75 

Now 30 cwt. is the additional weight that will break the first beam ; and 17 cwt. 3 lb. 
8 oz. the weight that will break the second : in which the reader will observe, that 10 :: 3 :: 8 
has a much less proportion to 20, than 20 cwt. 17 lb. 8 oz. has to 21 :: 56. From these ex- 
amples the reader may see that a proper allowance ought to be made for all horizontal beams ; 
that is, half the weights of beams ought to be deducted out of the whole weight that they will 
carry, and that will give the weight that each piece will bear. 

If several pieces of timber of the same scantling and length are applied one above 
another, and supported by props at each end, they will be no stronger than if they were laid 
side by side; or this, which is the same thing, the pieces that are applied one above 
another are no stronger than one single piece whose width is the width of the several pieces 
collected into one, and its depth the depth of one of the pieces ; it is therefore useless 
to cut a piece of timber lengthways, and apply the pieces so cut one above another, for these 
pieces are not so strong as before, even if bolted. 

EXAMPLE. 

Suppose a girder 16 inches deep, 12 inches thick, the length is immaterial, and let the 
depth be cut lengthways in two equal pieces ; then will each piece be 8 inches deep, 
and 12 inches thick. Now, according to the rule of proportioning timber, the square of 16 
inches, that is, the depth before it was cut, is 256, and the square of 8 inches is 64 ; but 
twice 64 is only 128, therefore it appears that the two pieces applied one above another is 
but half the strength of the solid piece, because 256 is double 128. 

If a girder be cut lengthways in a perpendicular direction, the ends turned contrary, 
and then bolted together, it will be but very little stronger than before it was cut ; for al- 
though the ends being turned give to the girder an equal strength throughout, yet wherever 
a bolt is, there it will be weaker, and it is very doubtful whether the girder will be any 
stronger for this process of sawing and bolting ; and I say this from experience. 

If there are two pieces of timber of an equal scantling (PI. 51, Fig. B,) the one lying 
horizontal, and the other inclined, the horizontal piece being supported at the points e and 
/, and the inclined piece at c and d, perpendicularly over e and/, according to the principles 

10* 



76 INTRODUCTION TO 

of mechanics, these pieces will he equally strong. But, to reason a little on this matter,. 
let it be considered, that although the inclined piece D is longer, yet the weight has less effect 
upon it when placed in the middle, than the weight at h has upon the horizontal piece C^ 
the weights being the same ; it is therefore reasonable to conclude, that in these positions 
the one will bear equal to the other. 

The foregoing rules will be found of excellent use when timber is wanted to support a 
great weight ; for, by knowing the superincumbent weight, the strength may be computed to 
a great degree of exactness, so that it shall be able to support the weight required. The 
consequence is as bad when there is too much timber, as when there is too little, for nothing 
is more requisite than a just proportion throughout the whole building, so that the strength 
of every part shall always be in proportion to the stress : for when there is more strength 
given to some pieces than others, it encumbers the building, and consequently the founda- 
tions are less capable of supporting the superstructure. 

No judicious person, who has the care of constructing buildings, should rely on tables of 
scantlings, such as are commonly in books ; for example, in story posts the scantlings, accord- 
ing to several authors, are as follows : - 

For 9 feet high 6 inches square. 

12 8 

15 10 

18 12 

Now, according to this table, the scantlings are increased in proportion to the height ; 
but there is no propriety in this, for each of these will bear weight in proportion to the num- 
bers 9, 16, 25, and 36, that is, in proportion to the square of their heights, 36 being 4 
times 9 ; therefore the piece that is 18 feet long, will bear four times as much weight as that 
piece which is 9 feet long ; but the 9 feet piece may have a much greater weight to carry 
than an 18 feet piece, suppose double : in this case it must be near 12 inches square instead 
of 6. The same is also to be observed in breast-summers, and in floors where they 
are wanted to support a great weight ; but in common buildings, where there are only 
customary weights to support, the common tables for floors will be near enough for 
practice. 



PRACTICAL CARPENTRY. 77 

To conclude the subject, it may be proper to notice the following observations which seve- 
ral authors have judiciously made, viz. that in all timber there is moisture, wherefore all bearing 
timber ought to have a moderate camber, or roundness on the upper side, for till that moisture 
is dried out the timber will swag with its own weight. 

But then observe, that it is best to truss girders when they are fresh sawn out, for by their 
drying and shrinking, the trusses become more and more tight. 

That all beams or ties be cut, or in framing forced to a roundness, such as an inch in twenty 
feet in length, and that principal rafters also be cut or forced in framing, as before ; because 
all joists, though ever so well framed, by the shrinking of the timber and weight of the covering 
will swag, sometimes so much as not only to be visible, but to offend the eye: by this precau- 
tion the truss will always appear well. 

Likewise observe, that all case bays either in floors or roofs do not exceed twelve feet 
if possible, that is, do not let your joints in floors exceed twelve feet, nor your purlines in 
roofe, &c. but rather let their bearing be eight, nine, or ten feet; this should be regarded in 
forming the plan. 

Also in bridging floors, do not place your binding or strong joists above three, four, or five 
feet apart, and that your bridging or common joists are not above ten or twelve inches apart, 
that is, between one joist and another. 

Also, in fitting down tie beams upon the wall plates, never make your cocking too large, 
nor yet too near the outside of the wall plate, for the grain of the wood being cut across in the 
tie beam, the piece that remains upon its end will be apt to split off", but keeping it near the 
inside will tend to secure it. See plate 33, at the bottom, where the dimensions are figured. 

Likewise observe, never to make double tenons for bearing uses, such as binding joists, 
common joists, or purlines ; for, in the first place, it very much weakens whatever you 
frame it into, and in the second place it is a rarity to have a draught to both tenons, that 
is, to draw both joints close ; for the pin in passing through both tenons, if there is a draught 



rs INTRODUCTION TO PRACTICAL CARPENTRY. 

in each, will bend so much, that unless it be as tough as wire, it must needs break in 
driving, and consequently do more hurt than good. 

Roofs will be much stronger if the purlines are notched above the principal rafters, than 
if they are framed into the side of the principals j for by this means, when any weight is ap- 
plied in the middle of the purline it cannot bend, being confined by the other rafters ; and if 
it do, the sides of the other rafters must needs bend along with it, consequently it has the 
strength of all the other rafters sideways added to it. 



l^/a/c . Jf). 




Wf 



K 



■^ 



V 




DESIGNS OF ROOFS 



PLATE XXXIX. 

A and B shows the method of trussing girders as is used by the greatest masters at this 
time. 

C is a horizontal section of -B. 

Z) is a section of the butment, by cutting across abinA. 

E and F shows the two sides of the king-bolt, at c in A, which is made with a wedge- way 
upon the top, so that it may force out the trusses upon the hutments. 

The cores of the trusses ought not to be let close into the groves of the side beams, but 
should be well secured at the ends and in the middle ; for suppose a weight to be laid upon 
the girder sufficient to'bend it, if the braces are tightly fitted they will bend along with the 
beams, and consequently be of no use ; but if they are not fitted close, the girder can never 
have a curviture to any sensible degree, for the side beams cannot bend without shortening 
their length, but in the act of bending the braces force upon the ends, and consequently act in 
opposition to the side beam. 

In tightening the truss work the head of your king-bolt ought to be greased, so that it may 
slide freely past the ends of the trusses ; proceed to turn the nut of the king-bolt, and another 
person to hit the head at c, with a mallet, which will make it start every time it is hit, and 
give fresh ease at every turning of the nut, so that you may camber the girder to any degree 
that you shall have occasion for, but generally not above an inch in twenty feet. 

J^oU. The sections 2), Ey and Fj are to one-eighth part of the real size. 



80 DESIGNS OF ROOFS. 



PLATE XL. 

This contains the most simple construction of roofs. 

Fig. a is calculated for a small building; at one end of the collar-beam is the Carpenters' 
boast, what they term a dove-tail tenon ; but I think rather a rule joint, as it is worked out to a 
centre. This roof will do for an extent of 20 or 25 feet. 

Fig. jB is a truss for a roof, the purlines to be notched upon the principal rafters, as will be 
described in the following plates of ledgment roofs ; this roof may be well calculated for an ex- 
tent of 30 or 35 feet, the height one-fourth of the span for slate covering. 

Fig. C is a simple construction of a roof, for the segment finish of a dome, which will an- 
swer to any of the above extents. 

■^ 

PLATE XLL 

Fig. .;2 is a roof for the purlines to be framed in, and the common rafters to come fair with 
the principals. 

Fig. J5 is a roof calculated for a greater extent than any of the foregoing roofs, and may 
•^well extend 50 or 60 feet. Here likewise is shown the connexion of the roof in the walls. 

Fig. C is a roof supported by two queen posts, instead of a king post, to give room for a pas- 
sage or any other conveniency in the roof. 



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^ DESIGNS OF ROOFS. 81 



PLATE XLII. 

This roof is calculated for a spao of seventy or eighty feet. 

You will observe in this, and the foregoing roofs, that the trusses are the same in num- 
ber as the purlines which they have to support ; for how absurd it is to give a roof more 
strength than necessary! but, on the other hand, the consequences will be dangerous if 
too weak. 

Fig. ./^ is a design of a roof for a theatre, which may extend from 80 to 90 feet. 

As it happens frequently in building that walls run across the roof, in such cases there will 
be little occasion for trussing the roof; then the purlines may be trussed, which will save one 
or two pair of principals, which is a considerable advantage. 

Fig. ^ is a roof of this kind, which shows the ends of the purlines, and C shows how 
to truss the parline. 

D, E, and F, are the methods of scarfing timber. 

PLATE XLIII. 

Is explained on the Plate. 

PLATE XLIV. 

Fig. .4 is a curb roof, with a door in the middle of the partition ; the beam a 6 to run 
quite across the roof. 

Fig. jB is a roof calculated for two rooms. 

Fig. C shows the method of framing a bridge floor. 

[11] 



S2 DESIGNS OF ROOFS. 



PLATE XLV. 

Fig. a is the design of an M roof, which is useful in some cases where the span is great, 
and no wall between, and the roof is required not to appear of a great height ; but this seldom 
happens in practice, for if there is any wall between the external walls, the roofs are in general 
made double, as is shown at figures B, C, and D, ^ 



PLATE XLVL 

Fig. .^ is a design for a church roof, the extent marked on the plate. 

Fig. B is a design of the same kind, but may be applied to an extent much greater. These 
two roofs, when finished, will be the same in every respect with the cylindro-cylindric arches 
described in plate 23 ; as the manner there shown of fixing the ribs will not be different in this, 
I refer the reader to the description of that plate. 

Fig. C is another design for a church roof, where the ceiling over the galleries is to finish 
level. 



PLATE XLVIL 

Fig. .;5 is a design for a domical roof; B shows the manner of framing the curb for it to 
s-tand upon, the section of the curb being also shown upon the bottom of ^^. ,ji. 

JV. B. This design is nearly the same as that constructed for the dome of the Pantheon, 
London, which was burnt down. 



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u 



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II II I! II 11 II II. 




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DESIGNS OF ROOFS. 8S 



PLATE XLVIII. 

Fig. Jl is another design for a domical roof; the bottom of it is made into a very narrow 
compass, in order to gain room within the dome. 

Figures B and C are designs for circular and elliptical trusses for bridges, &c. These 
trusses may also be applied to roofs where there is no cavity wanted within. 



PLATE XLIX. 

Fig. ^ is a design for a story post and breast-summers. 

Fig. ^ is a design for a bridge. C is a section across. D is part of the plan, which also 
shows the manner of fixing the piles. E shows half the plan of the bridgings. 



11* 



J'/(lf( .'/('. 




HAND-RAILING. 



THE principles of hand-railing here described are entirely new ; former writers having 
failed from not considering the two vertical sides of a hand-rail as being the portion of a cylin- 
dric or prismatic surface : for supposing such a prism or cylinder to be erected on the plan of 
the rail, any portion of the rail will be contained between two parallel sections ; and to form 
it out of the least stuff, one of the sections may be supposed to touch the rail in three points in 
a vertical surface passing through the middle of the rail and the plan, one point at each end, 
and one point at or near the middle ; the other section may touch it in one or two points, as it 
might happen. The business of hand-railing is to find the position of the section, and to trace 
it out according to that position. 

PLATE L. 

To draw the form of a Hand-rail. 

In fig. Jl make an equilateral triangle vwt upon its width, and divide it into five equal 
parts, and from one part on each side draw zs andyifj, then /, g and m are the centres, Im 
being made equal to Ig ; the centres are found the same for the other side. 

The Form of a Rail being given, to draw the Mitre Cap. 

Let the projection of the cap be three inches and a half, and make the distance of the 
inside circle from the outside circle the projection of the nose on each side of the rail, and 
draw the mitre n o and p o ; then continue parallel lines down to the mitre p o, put the foot of 
your compass in the centre of the cap, and circle the parallel lines round to a,, c, e, g, and i, and 
draw the ordinates ab, c d, ef, &c. and then prick the cap to the rail according to the letters. 



86 STAIR. CASING. 



How to draw the Form of the Cap for the Mitre to come to the Centre. 

It is only drawing the parallel lines from the rail to the mitre wherever it is, and circling 
them round to the ordinates; and so pricked from the rail, and the thing is done. 



PLATE LI. 

Plan and elevation of a dog-leg stair, with a half space. 

PLATE LIL 

Plan and elevation of a dog-leg stair, with winders in the turning. 

PLATE LIII. 

To draw the Scroll of a Hand-rail. 

In fig. A make a circle three inches and a half diameter, divide the diameter into three 
equal parts, and make a square in the centre of the eye to one of those parts, and divide each 
side of the square into six equal parts ; this square is shown in E, at the bottom, in full size 
for practice, and laid in the same position as the little square above, so that the centres may 
be more readily found, which are all marked in a regular position 5 the centre at 1 draws from 
i round to k, the centre at 2 draws from k to /, and the centre at 3 draws from / to m, &c. 
which will complete the outside revolution at a, with the centre c ; then set the thickness of 
the rail from a to/, and go the reverse way to draw the inside ; then the scroll will be com- 
pleted. 



T.' ./r./ii- llw r.niip ,'t' fJiis sliiir. 

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STAIR-CASING. Sf 



To draw the Curtail Step. 

Set the ballisters in their proper places on each quarter of the scroll in fig. A; the first 
ballister shows the return of the nosing round the step, the second ballister is placed at the 
beginning of the twist, and the third ballister a quarter distant, and straight with the front of 
the last riser: then set the projection of your nosing without, and draw it all round equally 
distant from the scroll, which will give the form of the curtail. 

To draw the Face Mould for squaring the Twist Part of the Scroll. 

You will observe here, that the joint is made at 3, 6, just to clear the side of the scroll ; 
draw ordinates across the scroll at discretion, to cut the line dh, abc being the pitch-board ; 
take notice that lines be drawn from 3 and 6 to meet d h, so that you may have the said points 
exact at 3 and 6 in your face mould ; then take the line d 6, and mark the places of the ordi- 
nates upon a rod, and transfer the divisions to c? 6 in B, then trace B, from fig. A, according 
to the marks. 

To find the falling Mould C. 

In C, a Z> c is the pitch-board ; the height is divided into six parts, to give the level of the 
scroll ; the distance ad'\% from the face of the riser to the beginning of the twist ; and the dis- 
tance from c? to A in C, is the stretch-out from a, the beginning of the twist round to h in fig. A; 
each being any point taken at discretion, more than the first quarter ; divide the fevel of the 
scroll, and the rake of the pitch-board into a like number of parts, and complete the top edge 
of the mould by intersecting lines, and the under edge parallel to it to the depth of the rail. 

How to find the parallel Thickness of Stuff for the Twist and Scroll. 

Take the compass round abc d.e, to 6, in fig. A, and stretch it out upon the base of the 
pitch-board from d to g: draw g h perpendicular to intersect with the top of the mould ; then 
draw the dotted line hf, parallel to the level of the scroll both ways ; then take the distance 



\ 



88 STAIR-CASING. 

6 1, in fig. A, that is, the length of the plan, for the twist part, and set it from c? to e in C, 
and draw ef perpendicular, to cut the parallel fh ; then draw a dotted line through /, pa- 
rallel to c b, the longest side of the pitch-board, which gives the thickness of stuff for the 
twist, about three inches and a half ; and the parallel line from / to the base, shows the 
thickness of the scroll. 

JSfote. The falling mould Z), for the outside, is found in the same manner as the other 
falling mould C 



PLATE LIV. 

As the method of getting a scroll out of a solid piece of wood, having the grain of the 
wood to run in the same direction with the rail, is far preferable to any of the other methods, 
with joints in them, being much stronger than any other scroll with one or two joints, and 
much more beautiful when executed, as no joint can be seen, and consequently no difference 
in the grain of the wood at the same place ; I shall here give a specimen,^ the method for 
describing a scroll being already given in the last plate ; and likewise the falling mould. 

How to find the Face Mould. 

Place your pitch-board, a be, in fig. D, as in the last plate ; then draw ordinates across 
the scroll at discretion, and take the length of the line d b, with its divisions on the longest 
side of the pitch-board, and lay it on c? 6 in i5 ; then the ordinates being drawn in E, it will 
be traced from fig. 2), as the letters direct. 

How to find the parallel Thickness of Stuff. 

Let ab che the pitch-board in F, and let the level of the scroll rise one-sixth, as in the 
last plate ; and from the end of the pitch-board at 6, set from b to d half the thickness of the 
ballister, to the inside ; then set from dtoe half the width of the rail, and draw the form of 
the? rail on the end at e, the point b being where the front of the riser oomes, then the point e 
will be the projection of the rail before it ; then draw a dotted line to touch the nose of the 
scroll, parallel with c 6, the longest side of the pitch-board ; then will the distance between 



Plate .54-. 



Fu; D. 





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Thir/rius.s of SlnlT //•/ I',,,. A. Th "iriuss rf SlnlT l',<r i;,j /J 



TIatc 55. 



Laiiilimj. tor i>s /?'«,.<• see 



Hi .1 /.-.v/ 





'* 



STAIR-CASING. 89 

this dotted line and the under tip of the scroll show the true thickness of stuff, which is nearly 
five inches and a half: but there is no oc(^sion for the thickness to come quite to the under 
side ; if it comes to the under side of the hollow, it will be quite sufficient, as a little bit 
glued under the hollow could not be discernible, and can do no hurt to the scroll, therefore 
a piece about four inches and a half will do. 

Fig. »/^ is a scroll of a smaller size, drawn in the same manner and with the same cen- 
tres as the others are, but with a centre less. The method of finding the raking mould and 
thickness of stuff is the same as fig. B. 



PLATE LV. 

Plan and Elevation of a Geometrical Stair, without winders in the turning. 

PLATE LVt 

TART 1. 

Figure 1. No. 1. Part of the plan of the rail for a stair consisting of eight winders 
round the semicircular part, and flyers below and above. 

No. 2. The elevation of the convex side of the semi-cylinder showing the winders. 

No. 3. The elevation of the concave side, with the delineation of the winders, drawn 
from the equal parts on the concave side of the plan. 

No. 4. The steps stretched out for the semi-circumference of the convex side of the 
rail \ A B C being the triangle of the winders, and ABE and CFG flyers, one below and 
the other above the winders in position. 

No. 5. Half of the winders stretched out for the concave circumference. 

\- [12] 



90 STAIR-CASING. 

To draw the falling mould PQRSTJVIMLUK,^o. 4, produce Z) ..5 and ^ C upwards 
to Hand Oj make ^ H and CO of any equal heights; join HO', draw HK parallel to 
.^ Ef and JV parallel to G C : therefore JV and G K will be parallel to each other : make 
OJVand Oil/ also HL and HK equal to AE or GC'. and describe two tanged curves 
JVIMandKUL, and K U L M IJV W\\\ be the under side of the falling mould, then the 
upper curve P Q ^ its' y being drawn parallel to a given distance will complete the falling 
mould. In the common run of business the distance between the under and upper edges of 
the falling mould is 2 inches, sometimes only 2* inches, the breadth of the rail 2J inches. 
Much in the same manner most falling moulds are to be formed, viz. by placing a flyer before 
and after the winders. In semicircular rails upon level landings and half spaces, the winders 
are reduced to a single step with a broad tread equal to the circumference of the rail, and the 
height, the common height of the flyers : also in quarter spaces, the quadrantal periphery of 
the rail is considered as the tread of a step, in addition to the other winders which are neces- 
sary in the other quarter. 



PLATE LVI. 

PART 2. 

AB No. 5 is the pitch of the steps or falling mould for the concave side of the rail, CD 
the pitch of the steps or of the falling mould for the convex side of the rail ; let these two 
pitches intersect one another in E, in the middle oi AB: it must be obvious that neither of 
these positions gives the true pitch of the rail piece. If the rail were considered in depth only 
without thickness, then the middle part of the cylindric projection, as shown in the elevation 
No. 2, would give the pitch, provided the radius of the rail were the same as the outside of 
the plan. No. 2. Therefore neither the outside nor inside falling moulds, nor the elevations of 
the concave or convex projections of the rail will give its pitch, but from what has now been 
shown of the projections of the cylinder it will be no difficult matter to form an idea of the 
complete projection of a rail piece, showing the entire solid in order to form the true pitch. 
This will be described in the next plate. 

Fig. 3. The manner of drawing a tanged curve from the formation of a parabola by 
the intersections of lines : thus, let BA and B C be two straight lines forming an angle, and 



riil 




Ph,/,' v: 





-1=. ^ 



STAIR-CASITsG. 91 

let it be required to tange these lines at Jl and C by a curve, divide AB and B C each into the 
same number of equal parts, or in the same proportion, then number the divisions oi AB from 
A towards B, and those of B C from B towards C, and through each two corresponding points 
draw a straight line, and all the straight lines drawn through each two such points will form a 
tanged curve, which is that required : by corresponding parts, as the first from A, and the first 
from B, as the extremities of the first line ; and from the second from A to the second from B, 
and so on. This is the method by which the rail is supposed to be eased K UL and MUST, 

Fig. 2. Another method of drawing a tanged curve by a circular arc. Fig. 1. No. 6, 
pitch-board of the flyers. Fig. 1. No. 7, pitch-board of the winders. 



PLATE LVII. 

PART 1. 

To show the proper twist of a rail and thickness of stuiF, also the face moulds for the same. 

Fig. 1. No. 1. AA'0'0 be the plan of a rail : figure 2, the falling mould as completed. 
Let the plan of the rail No. 1, fig. 1, be so placed that the chord of t/5 C be parallel to the 
base MO of the falling mould /^. 2, and let B B' be the separation of the straight a,nd circu- 
lar parts of the rail, 0' B' B being the quadrantal part of the rail, and B' A' A B the 
straight part : divide the concave side B C D oi the quadrantal part into equal parts at the 
points B, C, B, &c. ; through all the points B, C, B, &c. draw lines at right angles to the 
chord A O, cutting it in 1, 2, 3, &c. and produce them upwards to the points/, g, h, &c.; let 
MBJS% fig. 2, be the section of a step, upon MO make B A equal to B A, No. 1, figure 1 : 
extend the parts AB, B C, C D, &c. No. 1, figure 1, upon the base MO^ figure 2, from A 
to B, from B to C, from C to B, &c. : from the points A, B, C, D, &c. draw lines perpendi- 
cular to MO cutting the lower edge of the falling mould at E, F, G, H, &c. and the upper 
edge at /, J, K, L, &c. In figure 1, draw any line, abed, &c. parallel to the chord A ; 
make ae, bf, eg, dh, &c. respectively equal to AE, BF, C G, B H, &c. figure 2 ; also in 
figure 1, make ai, bj, ck, dl, &c. equal to AI, BJ, C K, D L, &c. : through the points e, 

12* 



92 STAIR-CASING. 

/, g^ h, &c. figure 1, draw a curve ; also through the points i,j, k, I, &c. draw another curve, 
and these two curves will complete the projection of the falling mould. From the point s, 
figure 1, radiate the lines, CC, DD'f &c. cutting the convex side at C B' , &c. : from the 
points ja, B', C, B', &c. draw the lines A' i', B'f, C h! , B' h', &c. : draw ee', ii',jj', kh!, 
hh', &c. parallel to the chord t^ : through the points i,jy k, &c. draw a curve until it inter- 
sect with the curve i,jf kj &c. this will form the projection of the top of the rail piece. In 
like manner the under parts which appear at q q', h h! , will be completed in the same manner, 
so that No. 2 is the whole appearance of the solid, qr' r q' and i i' e e' being sections or the 
ends which join the contiguous parts of the rail. 



PLATE LVII. 

% PART 2. 

To trace the face mould No. 3 ; join i' r, and let the perpendiculars cut i r at t, 1, 2, 3, &c. 
draw the ordinates lbu,2cVf^ dw, &c. perpendicular to ir'. make 1 6, 2 c, 3 d, &c. equal 
IB, 2 C, 3 2), &c. No. 1 : also in No. 3, 1 u, 2Vf 3 w, &c. equal to 1 U, also make x o 
equal to X 0', i a' equal to 7^', 2, V,^ W, and C : draw b y parallel to t a', and a y parallel 
to t b, and ta'yh will complete the straight part of the face mould : join r o 5 draw the con- 
cave curve b e d . . . .r, and the convex curve y uvw o, which completes the curve 

part and the whole of the face mould No. 3. 

Fig. 3 shows the projection and face mould for the quadrantal part of a rail, the principles 
of projection and manner of drawing the face mould is the same as what has now been shown. 
This figure is introduced to show how much less wood the part of the rail requires from a 
quadrantal plan, than when a straight part of the rail is taken in, and the more of the straight 
rail that is taken in, the greater will be the deflection from the chord ; thus 'va. figure 3, the dis- 
tance between the chord f r at any point to the nearest point of the projection, is much less 
than the distance i' r, fig. 1, from a corresponding point in i' r to the nearest point of the pro- 
jection. 

The dotted lines fig. 3, show another face mould, the chord being drawn from the one 
extreme point to the other, of the upper end of each section, which makes the face mould 



Flat^ .58. 





■ V W V 1( cj 




STAIR-CASING. 93 

much shorter than it ought to be, and will therefore require a much greater thickness of stuff. 
From this false position j r of the chord, some unskilful teachers of drawing direct their 
pupils to trace the face mould of a hand-rail. 



PLATE LVIII. 

To find the Face Mould of a Stair -case, so that ivhen set to its proper Rake it will be perpendi- 
cular to the Plan whereon it stands for a level Landing, as is shown in the last Plate. 

In fig. A draw the central line h q, parallel to the sides of the rail ; on the right line b q 
apply the pitch-board of a flyer; from q to a draw ordinates nd, me, If, kg, and i k, at 
discretion, taking care that one of the lines, as k g, touch the inside of the rail at the point g, 
so that you may obtain the same point exactly in the face mould ; then take the parts q u, uv, 
vw, w X, xy, and apply them at B from qu, uv, vw, w x, xy, from these points draw the 
ordinates of B, and prick them from the plan, figure Jl, as the letters explain ; then B will be 
the mould required. 

To find the Falling Mould. 

Divide the radius of the circle fig. A, into four equal parts, and set three of these parts 
from 4 to 6 j through n and v, the extremities of the diameter of the rail, draw b n and b v, 
to cut the tangent line at the points c and d ; then will c J be the circumference of the rail, 
which is applied from c to d, at C, as a base line : make c e the height of a step ; draw the 
hypothenuse e d, at the point e and d ; apply the pitch-board of a common step at each end 
of their bases, parallel to c d, make df equal to d e, if it will admit of it, and by these lengths 
curve off the angles by the common method of intersecting lines ; then draw a line parallel to 
it. for the upper edge of the mould. 



94 HAND-RAILING. 



PLATE LIX. 



To construct a Hand Rail with Butt Joints. 

Let No. 1 be the plan of the rail, consisting of the semicircular part and two straight 
parts, as each of the rail pieces or wreaths to be executed is to be carried over the corres- 
ponding straight part of the plan, as much of the plan requires to be shown as each wreath 
will cover. 

Draw ^ F, No. 2, parallel to the springing line B D of the plan, No. 1, and through the 
centre ^ of the circular part of No. 1, draw K L K, which will separate the quadrantal parts 
of the rail at / /, No. J, make the part IB oi F Jl, No. 2, equal to the stretch-out of the 
quadrant IB, No. 1 : the part I D oi A F, No. 2, equal to the stretch-out of the correspond- 
ing quadrant ID, No. 1, then complete the falling, No. 2, as in plate 56, A C being the hy- 
pothenuse of the adjoining flyer below the winders,- C E the hypothenuse of the winders, and 
E H the hypothenuse of the adjoining flyers above the winders. Let K K cut the lower 
edge of the falling mould at L, and the upper edge at K' . Make the part B O, No. 2, equal 
to B 0, No. 1, and draw O P perpendicular to A F, cutting the upper edge of the falling 
mould at P. In No. 2, bisect K' L at J: through t7"draw MJV* perpendicular-: to C E meet- 
ing C E a.t M, and the upper edge of the falling mould at JV. Through Mdraw JT F paral- 
lel to the middle vertical line K' K, cutting the upper edge of the falling mould No. 2, at X, 
and the base A F dX V: make I Von the convex side of the plan No. 1, equal to / Fon the 
base A F No. 2: radiate VIC No. 1, cutting the concave side at v' : bisect Vv' at v, and 
O O at o ; join o 37 and draw O A parallel to o x, cutting the concave side at A : from the 
centre .fi' radiate A Y, cutting the convex side of the plan at Y: draw op and v x' perpendi- 
cular to o 37 or A: make op equal to P No. 2 ; also make v x equal to F JTNo. 2 ; and 
draw p X, then upon p x as a, chord complete the face mould. No. 3 of the lower quarter of 
the winders, according to the plan 0' OBZRIVYAv'xrCB' 0', as in plate 58. 

Much in the same manner the face mould No. 4, of the upper quarter winders is found, as 
appears by the corresponding letters on No. 1 and No. 2. 




//////' .v/y. 



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Fig. A. 



V^ 



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HAND-RAILING. 95 

No. 5 is the edge of the plank elevated to its proper pitch, as required ; A F, No. 2, be- 
ing esteemed as the level whence it springs, the angle C B A being equal to the angle pxv. 
No. 1, and B C or AD, No. 5, equal toj&' x' at No. 3> that is, equal to the distance between 
the points of contact of the face mould and its chord. 

In like manner. No. 6 shows the edge of the plank for No. 4. 



PLATE LX. 

To draw the Falling Mould of a Rail having a Quarter Space in the Stair ; thence to find the 

Face Moulds of the circular Part. 

At the plan fig. A, ac'is, the stretch-out of half the circular part of the rail, found by the 
same method as in the foregoing plates ; or it may be found more exactly thus : divide the ra- 
dius into four equal parts, and set three of the divisions out to 3, and draw a line from 3 to b, 
cutting the side of the rail produced at a;* from the point/ in the right line kg sX B, make 
fh, and fg, each equal to the stretch-out of half the rail, that is, equal to a c, fig. A ; draw 
the perpendiculars ho, fl and gt: at -B apply the pitch-board of a common step at F', 
through the point t draw tk, parallel to gh, cutting the line //, at k ; from ^ to / set up the 
height of the four winders ; through I draw In, parallel to g h, cutting the line h o, at n ; from 
n make n o, equal to the height of a step, for the quarter space upon the landing, which only 
rises one step : draw the hypothenuse /o; again, draw op parallel to g h, and j& §- 5 perpendicu- 
lar to op draw q o x then op q is the pitch-board of another common step above the winders : 
then these angles being cut off by the method of intersecting lines, the falling mould will be 
completed, as in the last plate ; make fu and fv, from /, equal to a d, fig. A, that is, the 
stretch-out from the middle of the arc at b, to the joint; draw vx and uz parallel to//; then 
take the heights from 1 to y and z, and set them from a to b and c, will give the section b c : 

* The line a c is nearly equal to the semi-circumference, and is the most exact of any that ever has yet been 
shown by a geometrical method ; it may be depended on in practice ; it is absolutely impossible to find a right line 
exactly equal to the circumference of a circle ; this has exercised the attention of the greatest mathematicians in 
every age. 



96 HAND-RAILING. 

then take m I from the falling mould, and from d make d e equal to it, will give the section 
D E ; then take w x from B, and make f g, at E, equal to it ; from w draw w r, parallel to g h, 
cutting /m at r ; from r take the heights from m and I, and set up these heights from ky to 1 
and K at E, it will give the section i k ; then the face moulds upon D and E will be traced as 
directed in the last plate. 



PLATE LXI. 

Wants no explanation. 

PLATE LXIL 

To draw a falling Mould for a Rail having Winders all round the circular Part, as is shown 
in the last Plate, thence to find the Pace Mould. 

To describe every particular in this, would almost be repeating what has been already 
described in the last plate ; the heights are marked the same upon the falling mould at D, as 
they are at the face mould, which will give the heights of the sections of the rail ; and the face 
mould at C is traced from the plan B, according to the letters ; in plate 61 is shown the same 
thing, only with this difference, that the face mould is partly straight at one end , but the me- 
thod of tracing this is nothing different from the other in the last plate : only I would have 
the reader to observe, in plate 61, that ordinates are drawn through the places where the cir- 
cular part begins, which will give the same points on the face mould ; for, by this means, you 
will be able to determine what part of the face mould is exactly straight, and where the crook- 
ed place of your mould begins. I hope the reader will understand the same thing in the fol- 
lowing plates, without repeating it a second time. G shows the application of the mould to 
the plank ; take the bevel at IT, and apply it to the edge of the plank at D, and draw the line 
b c ; then apply your mould to the top of the plank, keeping one corner of it to the point b, 
and the other corner close to the same edge of the plank ; then draw the top face of the plank 
by your mould ; then take your mould, and apply it to the under side at c, in the same 
manner. 



■,i 



VI ah 01 



f'l^i J/,, 




//./// 




I 






%;^ 



# 



r/n/r (O. 




Flal^ (j4. 




STAIR-CASING. 97 



PLATE LXIII. 

Plan and elevation of an elliptical stair-case. In every kind of stair-case whatever, the 
breadths of the heads of the steps are always reckoned on a line, bisecting their length, or at 
18 inches distant from the rail. In this example the steps are divided into equal parts, both 
at the rail and at the wall. This division will make the falling mould straight on the edges, 
and consequently will form an easy skirting as well as an easy rail. 



PLATE LXIV. 

How to draw the Face Moulds of an elliptic Stair. 

The plan and section being laid down as in plate 63, the reader will observe, that the 
ends of the steps are equally divided at each end ; that is, they are equally divided round the 
elliptic wall, and also at the rail. In this plate, the rail is laid down to a larger size than 
that in the last plate : the plan of this rail must be divided round, into as many equal parts as 
there are steps ; then take the treads of as many steps as you please, suppose 8, and let h h 
3Xfig. H, be the tread of 8 steps from H; on the perpendicular h m set up the height of as 
many steps, that is, 8 ; and draw the hypothenuse m h, which will give the under edge of the 
falling mould. The reader will observe, that this falling mould will be a straight line, ex- 
cepting a little turn at the landing and at the scroll, where the rail must have a little bend at 
these places, in order to bring it level to the landing and to the scroll ; then mark the plan 
of your rail in as many places as you would have pieces in yOur rail (in this plan are three) ; 
then draw a chord line for each piece to the joints ; also draw lines parallel to the chords, to 
touch the convex side of the plan of the rail ; from every joint draw perpendiculars to their 
respective chords. Now the tread of the middle piece at C being just 8 steps, the height 
of the section from A to m is 8 steps ; and the section mn \s the same as w w on the falling 
mould, and the section h i is the same height as h i upon the falling mould ; draw a line to 
touch the sections, and complete your face mould as in the foregoing plates : in each of the 
other pieces at E and G, the number of treads being 6 ; therefore, from your falling mould 

[13] 



98 STAIR-CASING. 

set the stretch of six steps ; from h to h draw h I, parallel to h n, then nkl will give the 
height of the sections at D and E : every thing else agreeably to the letters. 

JVote. The stretch-out of 8 steps, or any other number, is not reckoned on the chord ; 
but it is the stretch-out round the convex side of the rail, or what most people call the 
inside. 



PLATE LXV. 

PART 1. 

To find the Moulds for executing a Rail with a Semicircle of Winders. 

Figure 1. The method of laying out the falling mould as described in plate 56, except 
that the middle part does not follow the line of nosings, but is raised six inches, as is the 
practice with several handrailers, in order that the rail should not approach nearer to the 
nosings of the winders than to the flyers. This example is adapted to a stair with ten 
winders in the semi-circumference. 

Fig. 2. The plan and face mould of the lower quarter winders. 

Fig. 3. The plan and face mould of the upper quarter winders. ABC the convex 
side of the quadrantal part of the plan, C D a. straight part intended to be wrought on the 
same piece with the twisted part. In this method the plane of the top of the face mould is 
supposed to rest upon the upper extremities of three straight lines or slender rods perpen- 
dicular to the plane of the seat of the said face, and these three perpendicular lines to rise from 
three points, in a line dividing the breadth of the plan every where into two equal parts of 
the rail piece ; and these three points to be so situated that one may be at each extremity, 
one at the end of the quadrant, and one at the end of the straight piece, and the intermediate 
point at the intersection of a perpendicular drawn from the centre of the plan of the rail 
piece to a straight line joining the two extreme points. Let each of the upper extremities 
of the three perpendicular lines be called resting points, and let the feet of the perpendicu- 
lars be called the foot of the heights of the rail piece, which will therefore be the same as 



jPhi/e o\ 




STAIR-CASING. 99 

the seats of the resting points ; and let the three perpendicular lines themselves be called 
the heights of the rail piece, and their places distinguished by the lower height, the middle 
height, and the upper height. 

For fig. 2, let a be the foot of the upper height, h the foot of the middle height, and d 
the foot of the lower height ; join « c?; draw af, b' g and dh perpendicular to a d, the middle 
one h g being drawn from the centre E. Let A, B, C, D, figure 1, be the points corres- 
ponding to t/2, B, Cf J9, in the convex side of the plan, figure 2, and let .^ jP, B G and 
Z) H, be the heights of the rail piece, ^^wre 1 ; make a/, b' g, dh, figure 2, respectively equal 
to A F, B G, D H, figure 1 : pin/h, figure 2 : draw^ ^ parallel' to b d, cutting/ A at i : draw 
i k parallel to g b, cutting b d at k: draw ^ / m perpendicular to /A, cutting/^ at /; join b k ; 
from i, with the distance b k describe an arc at m \ join i m : draw B' P parallel to da from 
the extremity D of the concave side j produce the convex quadrant C B j1 to Q, and the con- 
cave quadrant C B' A to P, and radiate the line P Q from the centre JE, which will complete 
the whole plan of the rail piece ; the part P A' A Q, will make a sufficient allowance for 
the cutting of the joint. Draw ordinates parallel to bk cutting the chord B' P, the concave 
side of the plan, and the convex side of the same : produce bkto meet D' P in t, and produce 
m i to M, making i u equal tokt\ through u draw v w parallel to/A : from the points where 



PLATE LXV. 

PART 2. 

the ordinates intersect D' P, draw lines parallel to af, b g, or d h, cutting v w : from 
the cutting points invw draw lines parallel to um as ordinates : transfer the interior ordi- 
nates from the plan to the face mould, also transfer the exterior ordinates of the plan to the 
face mould, applying them from the intersected points in the chord v w, and through the 
points thus set off, trace the concave and convex curves, as also the straight part of the 
mould ; observe, however, that as the straight part of the mould is a parallelogram, that if 
three points on two contiguous sides are found, joining the middle point to each of the 
other two gives two sides ; each of the other two remaining sides is found by drawing a line 
parallel to its opposite side. 

13* 



100 STAIR-GASING. 

In the same manner the mould figure 3, is to be found; but the base line of the heights 
is taken upon any convenient line S R, parallel to ^ B, fig. 1, so as to shorten the height 
lines, as otherwise figure 3 would occupy more space than might be found at all times con- 
venient, and at any rate the shortening of the heights will shorten the time of drawing 
figure 3, as shorter lines can be drawn sooner than longer ones. The distance between the 
height lines of the upper rail piece is the same as those for the under rail piece. 

To find the spring of the plank fig. 4 : draw any straight line A B ; from which cut off 
B d equal to g I, figured : draw d C, figure 4, perpendicular io A B'. make d C equal to 
b b' figure 2, and join B C, figure 4, then the angle A B C is, denominated the spring of the 
plank, and the angle is said to be acute when the planes of the top and edge of the plank 
form an acute angle with each other : but when these two form an obtuse angle with each 
other the spring is said to be obtuse. 

To find the Spring of the Plane at an Obtuse Angle. 

\n figure 5, let A B he any straight line, which produce to d: make B d equal to B d 
figure 3 : in figure 5 draw d C perpendicular to^ ^ : make d C equal to a e, fig we 3, then 
ABC will be the spring of the plank at the obtuse angle. 

Both these bevels are supposed to be applied from the top of the plank : but if the com- 
plimental angle of the obtuse spring bevel which answers to the upper wreath piece be 
taken, then the lower spring is applied from the top in order to give the spring of the lower 
wreath piece, and the complimental spring of the upper wreathed piece to the lower edge of 
the said piece. 

The reader will perceive that in figures 2 and 3, though the plan is the same in both, 
but their position inverted, the face mould of the lower wreath piece is much longer than that 
of the upper one. This circumstance is owing to the middle parts of the falling mould being 
raised over the nosings of the winders, and the more it is raised above the winders the greater 
will the face mould of the lower wreathed piece exceed the length of the mould of the 
upper wreathed piece, and unless that (if supposing a line passing through the middle of 
the falling mould bisecting the breadth of the same) the distance of the line thus passing be 
the same over the winders as over the flyers, the two face moulds can never be equal. 



/ 



/'' ■' a I 




STAIR-CASING. fll 

PLATE LXVI. 

To describe the Moulds for a Hand Rail, ivhen the Plank is sprung with Butt Joints. 

Fig. 1. Plan of the rail with the lower and upper face moulds: No. 1, the plan of the 
lower wreathed rail piece: No. 2, the face mould of the said rail, piece : No. 3, plan of the 
upper wreath : No. 4, face mould of the same. 

Fig. 2. No. 1. Falling mould described as in plate 56 ; the extreme heights shown upon 
this diagram are found, and the plan^^. 1, prepared in the same manner as in plate 59. The 
face moulds are traced in the same manner as in the last plate for spliced joints. 

Fig. 3. The spring of the plank for the lower wreathed piece of the rail. 

Fig. 4. The spring of the plank for the upper wreathed piece of the rail. 

Fig. 5. A section of the plank for the lower wreathed piece bevelled to the spring, 
figure 3. 

Fig. 6. A section of the plank for the upper wreathed piece bevelled to the spring, 
figure 4. 

To form the wreath in the solid ; first spring the edge of the plank, then suppose the plank 
to be set in its position, draw the plumb bevel, say at the upper end ; apply the face mould to 
tlie top of the plank, so that the upper end of the convex side may Coincide with the draught 
at the arris, and bring the lower point of the face mould on the inner or concave edge close to 
the upper arris at the lower end, then draw the form of the mould on the plank ; invert the 
faces of the plank, keeping in mind which end is to be uppermost ; place the mould again, so 
that its top may be next to the top of the plank, and the upper end of the concave side of the 
mould close to the draught at the arris", as also the lower end of the inner side upon the arris ; 
in this position draw the form of the mould on the plank. In cutting out the plank the line of 
teeth of the saw should every where make the same angle as the plumb bevel drawn on the 
edge. 



102 HAND-RAILING. 



PLATE LXVII. 

Given the outside or inside falling Mould of a Hand Rail round a level landing, and the plan 
of the rail, to trace the face mould for either quarter of the landing, so that the rail may 
rest upon three points in the middle of its thickness, corresponding in height and in angular 
situation with respect to the axis of the well hole to the three points of the falling mould. 

l^tt^B CDS FA be the falling mould, figure 2, for the first half of the level land- 
ing; FGHthe pitch-board of a flyer; K L M JST O P K, figure I, the plan of the circu- 
lar part : L P KL, figure 2, the corresponding quarter to the falling mould : Q the centre 
of the plan ; c, T, d, three points in the middle of the rail : FJihe. rectification of the arc 
Ki L on the plan : JD half the height of the step : IB cutting the rail at E and B, bisect- 
ing i^/ perpendicularly, the height of the quarter of the rail in the middle; FA height at 
the lowest end, and JC at the highest : bisect KP and L aX c and d, and draw the chord 
cd, perpendicular to which draw cfa, Q Ui b, dj C. Draw fij parallel to c d, touching 
the plan of the rail at i : make/« equal to i^t^ ; ib equal IB, j c equal JC, and join a c : 
draw b R parallel to fj or c d, cutting ac aX R: draw R S parallel to b Q, cutting cd at S, 
and join ST', then *S'Tis the directing ordinate of the base : draw Zb y perpendicular to 
a c, cutting it at Z. In the right angled triangle V WX make V fF equal T U, the per- 
pendicular WX equal bZ, and join VX: make Z F equal VX; join YR; then YR is 
the directing ordinate of the section. Draw ordinates parallel to S T, cutting the chord c d, 
and the concave and convex side of the rail ; from the intersections of the ordinates with c d, 
draw lines parallel to f? c or c a, cutting a c : from the points of intersection in a c, draw in- 
definite ordinates parallel to R Y: transfer the intercepted ordinates of the plan between cd 
and the concave side from- a c, upon the indefinite ordinates, through the extremities draw a 
curve, which will be the concave side of the rail : in the same manner transfer the inter- 
cepted ordinates of the plan, contained between c d and the exterior side of the curve, on the 
indefinite ordinates of the section from a c, and through the points thus transferred draw a 
curve, which will give the convex side of the falling mould. Join P, which will be paral- 
lel to dc: draw G H parallel to T S, cutting P at H: draw G g parallel to d C, cutting 
aC m g: draw g h parallel to YR : make g h equal to G H; through h draw p o parallel to 
a C : draw Pp and o parallel to d C, then p and o will give the extremities of the concave 
sides of the mould. The extremities of the convex side of the mould are found by drawing 



4 



J^lxk in 





J'lafe 68. 




XiiliB/ 



HAND-RAILING. 103 

an ordinate on the plan through each extremity of the convex side, and finding the correspond- 
ing points of the face mould. 

VXZ, figure 3, is the "spring of the plank. 

The completion of the upper half of the face mould for the wreath part of the rail will be 
shown in the next plate. 



PLATE LXVIII. 



Fig. 1. The upper part of the falling mould in the last plate. 

Fig. 2. The face mould for the upper quadrantal^part of the rail ; the heights ab, cd, ef, 
are equal to the heights .>5^, CD,EF, of the falling mould,.j?^Mre 1. It must be observed, 
that as the lower part of the falling mould, plate 67, is convex on the top edge, and that at 
figure 1, exhibited in this plate, concave from the same side, the acute angle is formed by the 
chord, and the ordinates of the face mould are downwards in plate 67, and upwards in plate 68 : 
the former is the section of the segment of a cylinder making an obtuse angle with the plane 
of the segment, and the latter the section of the segment of a cylinder making an acute angle 
with the plane of the segment. In each of these cases, the plane of the segment is a plane 
standing perpendicularly upon the chord P, to the plane of the plan, plate 67, and upon 
A B, figure 2, perpendicular to its plane. The angle made by the plane of the segment and 
the plane of the plank, is termed the spring of the plank : therefore in plate 67, the spring of 
the plank will be an obtuse angle, and in this plate, an acute angle, which is shown at jIB C, 
figure 3, and thus formed, make B d equal to dn, fig. 2, draw dC aX right angles to ABf 
make dc equal to c m, and join B C, then A B C \s the spring of the plank. 

JV, B. The spring bevel is applied to the upper arris of the plank, whether the angle be 
obtuse or acute, with one leg upon the top and the other upon the edge, and with both legs 
perpendicular to the arris. Figure 4, shows the face moulds when the plank has no spring. 



104 HANDRAILING. 



PLATE LXIX. 

To form the Line JVosings of the Steps of a semicircular Winding Stair-case into a regular 
Curve of contrary Flexure; thence to find the Plan of the Steps, the Form of the String, 
and the Rail Stretched out. Fig. 1. 

Let Jl B C D F he a. part of the rail, BCD the semicircular part, ,j1 B and BE straight 
parts, each equal to the breadth of a flyer: let A I he the seat of the last riser. Draw B F 
parallel to .^ /, and make 5 i^ equal to 18 inches; describe a semi-circumference\F G^ ^: 
divide the breadth of the semi-circumference into equal chords, each equal to the breadth of a 
step, or as nearly equal to the breadth of a step as a semicircular arc will divide. The number 
thus contained in this example is nine: produce AB to JV; make ^ A* equal to half the 
stretch-out of the semicircular convexity 5 C J5 of the rail.' Draw JVO perpendicular to 
BJV\ make JVO equal to half the number of steps in the semi- circumference which are here 
4 1 : draw a 1 parallel to A .TV, make B K equal to the height of a step ; join .^^ K: produce 
»5 Z" cutting .A^O at ikf; bisect .^ ilf at L: join Z : makei^ equal io LA\ describe an 
arc to tange the straight lines L A and Lt dX A and t : parallel to JVA draw la, 2 b, 3 c, 4 d,. 
5 e, cutting the curve at a 6 c: draw «/, J ^, ch, di,ek, cutting .^JV at /j g-. A, i,k: trans- 
fer the distances/^, gh, hi, ik, kJS/', to the arc jB! CD, from B to /, from / to in, from m to 
n, from n to o, from o to C : make Cp, C q, Cr^ &c. equal to Co, Cn, C m, &c. aiid through 
the points F, 1, 2, 3, &c. draw/.Fj g P, m Q, n B, &c. which will represent the winders; 
theii AabedeO is half the line of nosings which may be transferred to figure 2 zX Aab cdef 
&c. thence the form of the steps and string are drawn ; the under edge of the falling mould, 
figure 3, is also of the same form, and the upper edge is drawn parallel to the under efdge. 

This formation of the string board is certainly a very great improvement in stairrcasing, 
as it gives one universal curvature^ to the soffit, that is, without any breaking of surface so as 
to form an angle, as is the case with all stair-cases with winders when the ends of the steps 
at the rail are equal, and where the breadths of the threads next to the rail differ from the fly- 
ers. The angle at the junction of the flyers and winders increases as this difference is greater. 



F7 ,te 69. 




ib^ 



M 



FhiU 70. 




STAIR-CASING. 105 



PLATE LXX. 

To diminish the Step of a Stair winding round one of the Quarters to a level Landing. 

Find the stretch-out round half the circular part of the rail, as directed for the foregoing 
plates, and complete the falling mould as directed in the last plate, for the winding part of the 
rail, which is six steps from t to y; in order, to bring the rail with an easy turn yound to the 
landing, set off the height of another step from u to 7, and let the under edge of the rail be 
half the height of a step above that to c ; or it may be more, according to the discretion of 
the workman ; then the rail will be half the height of a step more upon the landing than it is 
upon the winders ; through c draw c/ parallel to the base, and continue the line 2K, that 
forms the intersection below for the winders up to d, and cut off the angle m d c by intersect- 
ing lines, will give the under edge of the mould turning up to the landing : in order that the 
last step beyond the quarter should also follow the mould, draw a line through 7, the height 
of the last step, parallel to w 6, or c c, cutting the under side of the falling mould at a ; 
through A draw a b parallel to c ^ ; then u B is, the tread of the last step of the rail, which is 
set from ^ to e. The face moulds at D and F are completed in the same manner as directed 
in the last plate, and the moulds in plate 58, a.tfig. -Z), is also laid down by the same method, 
the height of the sections being taken from the falling mould that corresponds to that place of 
the rail which the face mould is made for 5 and the bevels that are laid down above each face 
mould will show how much you must bevel the edge of your plank, before you can apply the 
face moulds to the plank ; then draw the plumb of your rail, upon the bevelled edge, by the 
other bevels that are shown at the sections ; then apply your mould to each side of the plank, 
keeping it fair with the bevelled edge, the same as in other cases before mentioned. 

The reader who understands the foregoing examples will easily comprehend those 
at figure C in this plate ; for this purpose a mere inspection of the diagram will be 
sufficient. 

. [14] 



106 STAIR-CASINa. 



PLATE LXXI. 

This plate shows the method of capping an iron rail, upon much the same principles as the 
others, but with less trouble. 

How to find the Thickness of Stuff for capping of an Iron Rail. 

Lay a thin broad piece of wood, as 6, upon the top of the iron, upon the place that is to 
be capped, and turn it round upon the iron, till you see the greatest space between the iron 
and wood to be as little as possible ; then the open space will show what thickness must be add- 
ed to the thickness of the rail. 

How to find the Plumb of the Piece, for the Application of your Mould. 

After having found what thickness of stuff will do, apply the solid piece itself, b, to its 
place, then let one of the ends, as d e, be cut plumb. 

How to trace the under Side of the Plank to the Iron, instead of the Face Mould. 

Make a pricker, as c, with a steel point in the upper end, and let it be notched out, so that 
when it is applied on each side of the ballister, it may just leave the thickness of the rail 
between each point; then take your pricker c, and prick your piece b, at every ballister; 
always keeping your pricker close to every ballister. This being done both outside and 
inside, if you inspect C, it will make it plain, where you see the sides of the plank flatted 
out, which shows upon the under side at the black dots ; strike a plumb-line de, upon the 
end of your plank b, and this plumb-line will show how the top is to be pricked off from the 
bottom, which you see at C : the under side is squared over to the edge, from your pricked 
points, and from thence draw across the edge to the rake, which is formed by the plumb 
upon the edge, then squared over the top side, and then it is to be pricked off from a line 
drawn from the point d, in the end section D, which the; plumb-line gives upon the end, 
along the top face of the plank, parallel to the edge, and not from the square edge of the 
plank. 



rial, 7/ 



/'///.«■ I'l,il( .^lu'iiw f//,- ////////',/ /'/' ii////r///t/ 
ii// ircii III}/ 




This Shows Tuvr t^y gbu- a Rail in T7uam,ss. 



riifM 72 




STAIR-CASING. 107 



PLATE LXXII. 

This plate shows the method of glueing a rail in thicknesses ; but if I must give my opinion 
a rail got out of the solid is much preferable ; although you are obliged to have more end joints 
in it ; but if your joints are well screwed together, a solid rail has a more beautiful appearance 
than a rail glued up, having so many diiferent thicknesses of glue, which makes it have a black 
and nasty look with it; if a person is ever so careful, the joints will still show, and this rail in 
itself having a natural tendency to spring, the least dampness will make it give way in time ; 
but as this is held by some a very valuable acquisition, I shall proceed to lay down the moulds 
for it. 

To show the .application of the outside and inside falling Moulds to the upper and under Faces 
of the Plank, to give it the Form of the Twist. 

a b is the stretch-out of the greatest circle in B, and a c the height of the steps ; again, de 
is the compass of the lesser circle, set in the middle between a C and df, the height of the 
steps : therefore the triangle a i c is the pitch-board of the inside of the falling mould ; and bmo 
at the bottom, and i he a.t the top, are the pitch-boards of two flyers ; which lines, when inter- 
sected, will give the under line of the inside falling mould. In the same manner dfe, with the 
two flyers k gf at the top, and e In a.t the bottom, will give the under line of the outside falling 
mould. The top lines are only drawn parallel to the under lines to the thickness of the rail. 

How to apply these Moulds to the Plank. 

Draw a line tp to touch the moulds so laid down in two places, and as both moulds 
intersect together at q ; then draw a square line p q, upon the top of your plank, at the 
same distance as j& 5^ is from the bottom ends of your moulds, and this line being squared 
across the edge, and from thence across the under side : then set the distance p q on both 
sides of your plank, from the same edge, and likewise square over tsr, at the distance 
pt on your plank, on both sides, then set the distance of r from t upon the top side of 

14* 



108 STAIR-CASING. 

your plank, and set the distance oi st upon the under side ; you will observe to mark the 
point q upon both your moulds, then apply your outside falling mould to the top of your plank, 
making the plank q to coincide with the same point q in the plank, and make the front of the 
falling mould to come to r, and with this mould, placed in this position, draw the upper face of 
your plank with it, and in like manner apply your inside falling mould, that is, by applying the 
point q to the same point q in the plank, on the under side, and let the front edge be to s in 
the plank ; then draw the under side ; your plank being supposed of a sufBcient thickness, 
making allowance for the saw cuts, and plaining up your veneers : this plank, when cut out, 
twisted to those lines, will be the true form of your veneers : the piece being thus formed, you 
are to cut your veneers the other way into thicknesses as you think they will bend easy, and so 
I shall leave you to complete the rail. 

As to the height of the hand-rail, it must be left always to the discretion of the workman 
or architect, and though I have in these examples made the rail to follow the nosings of the 
steps, the principles of the face mould will still apply the same : for the falling mould may al- 
ways be supposed to be given. 



PLATE LXXIII. 

This plate shows the method of fitting down the skirting, upon any sort of stair-case what- 
ever 5 whether straight, circular, regular, or irregular j if the treads are ever so crooked, and 
the risers out of an upright. 

In Jig. *d, is shown a bevel, -made to the rake of the skirting, and the other perpendicular 
to the stair, and a sliding piece to be applied to the perpendicular side of the bevel with a 
hooked point of iron or steel, to stand forward at the bottom so much, that the sliding piece 
may clear the nosing of the step. I shall proceed to show its application. 



n,/rp 



This S/hni'S l/h- Mtlhi'J <'/ S/rir/iiuii/f /iN>S/(iir fds 



tl<l.-\. 







4 



STAIR-CASING. 109 



How to Jit down the Skirting. 

Lay the skirting over the top of the steps, and let a very fine notch be made on the front 
edge of your sliding piece, to the height of a step, or rather higher ; then apply the point of 
tlie sliding piece to the internal corner of a step, and prick your skirting in the notch at b, the 
bevel being supposed to be brought close to the slider : again, supposing you want to take a 
point at the nosing, where you see the bevel applied under, apply the point of your sliding 
piece to the nosing at c ; then prick your skirting in the notch at d, that will give the point d, 
w^hich is to correspond ^^^th c, &c. and by this means you may take as many pricks as will be 
sufficient, till the whole is completed. 

COROLLARY. 

- Hence it is evident by the same method, that one thing may be fitted into another, 
whether considered as a stair-case or not, standing either raking, horizontal, or perpen- 
dicular. 

If the steps of a stair- case be very true, two pricks from each riser and a tread will 
be sufiicient, as it is only joining these pricks by lines, which will form the rise and tread 
of each step, and three pricks in each nose, because a circle may be easily drawn through the 
three points. 

If the nosings are all exact, let a mould be made to fit one of them, and your nosings on the 
skirtings be drawn by this mould, which will likewise be exact. 

Fig. E shows the method of laying down a raking sided stair, which is clear in itself, the 
height of the steps being the same on each side. 

C and D shows the method of tracing one bracket from another, in a stair-case : C being 
the brackei for the common step, D a bracket for one of the winders. 

There is shown a method in the last plate, at E and D, for doing the same thing 
by means of a triangle, which is performed thus : let the other bracket at Jig. D be given 



.'C^ 



110 DIMINISHING RULES FOR COLUMNS. 

whose length is a b ; and if you want a bracket for the winders, whose length is b c, draw 
B c, making any angle, at the point -B join a c ; take as many ordinates as you please to 
touch all the principal lines of the given bracket ; then draw lines parallel to A c, from these 
ordinates ; and complete the other bracket as you see by the letters. 



PLATE LXXIV. 



How to diminish the Shaft of a Column by the ancient Method. 

In fig. A, describe a semicircle at the bottom ; let a line be drawn through the diameter 
at the top, parallel with the axis of the column, till it intersects the semicircle at 1, at the 
bottom; then 1, 1 at the bottom will be equal to 1, 1 at the top; divide the arch into four 
equal parts, and through these points draw lines parallel to the base, the height of the column 
being also divided into the same number of parts, and lines drawn parallel to the base, then 
the column is to be traced from the semicircle, according to the figures. 

How to diminish the Column by Lines drawn from a Centre at a Distance. 

Fig. C. Take the diameter a b, at the bottom, set the foot of your compass in c at the 
top, and cross the axis in the point d, continue c d aX the top, and « & at the bottom, to meet 
at . e ; then draw from c as many lines across the column as you please, and take the diameter 
a 6 at the bottom, and prick each line upon the axis equal to b a, which will give the swell of 
the column. 

To diminish a Column by Laths, upon the same Principle. 

In fig. D, the point e being found, as in fig. C, take and plow a rod d b, and lay the groove 
upon the axis of the column, and plow the describing rod upon the under side, and lay the 
groove upon a pin fixed at e, and fix a pin at g, to run in the groove upon the axis of the co- 
lumn, and the distance of the pencil at/, equal to b a, then move the pencil at/, it will de- 
scribe the diminishing. 



/7^///' 74. 





m 






•s 

r. 



Hat^ 73. 





'jffoi 'Ids/ar i) ' I't I u) fhi " /'V 'f'l of //' o ^ 



CTLINDRO-CYLINDRIC SASH-WORK, &c. ill 



How to deso'ibe the Column by another Method. 

Take the semi- diameter a 5 at bottom, and set the foot of your compass in the top at 
c, and cross the axis at 8, and draw the line a 8 on the outside, parallel to 6 8 on the axis, and 
di\-ide each of these lines into eight equal parts, and set the diameter a 6 at the bottom along 
the slant lines 1 1, 2 2, 3 3, &c. from the axis ; this will also give the diminishing of the 
column. 

How to make a diminishing Rule. 

DiN^ide the height of your rule into any number of equal parts, as 6 ; draw lines at right 
angles from these points across the rule, and divide the projection of the rule at the top ; that 
is, half of what the column diminishes , into the same number of equal parts put a pin or 
brad-awl ; lay a ruler from a to 5 ; mark the cross line at/; then lay a ruler from 4 to a, and 
mark the next cross line at e ; then lay the ruler from 3 to a, mark the next at d, and so on to 
the bottom ; bend a slip round these points, and draw the curve by it, will give a proper 
curve for the side of the column. 

J^''ote. This is the readiest method, and gives the best curve of any that I have 
tried. 



PLATE LXXV. 

The Plan and Elevation of a ciJ'cular Sash, in a circular Wall, being given ; to find the mould 
for the radial Bars, so that they shall be perpendicular to the Plan. 

Draw perpendiculars from the points 1, 1, 1, 1, &c. at A and B, in the radial bars, either 
equally divided, or taken at discretion, down upon the plan, to 1,2, 3, 4, 5, 6, 7, at C and D ; 
and draw a line from the first division upon the convex side parallel to the base ; then draw 
ordinates from 1, 1, 1, 1, &c. at right angles to the radial bars, at A and B, which being 



112 CYLINDRO-CYLINDRIC SASH-WORK, &c. 

pricked from the plans at D and C, will give a mould for each bar ; and the bevels upon the 
end will show the application of the moulds. 



To find the Veneer of the Arch-bar, or what is improperly called by Workmen Cot or 

Cod-bar. 

To avoid confusion, I have laid down the plan and elevation for the head of the sash 
under. The stretch-out of the veneer is got round 1, 2, 3, 4, 5, 6, on the arch-bar, 
which being pricked from the small distance on the plan at M, will give the veneer above, 
oXE. 

To find the Face-mould for the Sash-head. 

Divide the sash- head round, into any number of equal parts, at G, and draw them perpen- 
dicular to the base at H', draw the chord of one half of the plan at H, and draw a line paral- 
lel to it to touch the plan upon the back side ; then the distance between these lines at H, will 
show what thickness of stuff the head is to be made out of; and from the intersecting points 
on the back side, draw perpendiculars from the base of the face mould, which being pricked 
from the elevation, as the figures direct, will give the face mould. 

To find the Moulds for giving the Form of the Head, perpendicular to the Plan. 

The base of L is got round the arch 12 3 4 5 6, at jP, and the base of K is got round 
abed efg, also at F, and the heights of the ordinates of each are pricked either from Hot I, 
which will give both moulds. 

By the same method, a circular architrave, in a circular wall, may be got out of the 
solid. 

JVote. The face mould at G must be applied in the same manner as in groins; so 
that the sash-head must be bevelled by shifting the mould G, on each side, before you can 
apply the moulds K and L ; the black lines at JT and L are pricked from the plan, at 



m. 



riatc 70. 

An Arc ?n trove or Archivp/t fi>r ti C(/r//////- Wi/idcw in a Cirvu/ar W<i N 




T/iy firx/ ivnrff fj- he juirdy cii/ out "f /h,: Selid 
^'J /■(■// //rnye , s/^ Ibr ns li i cmd. Ic, f on .(/r/i -syd-etr ini 
Ti' Ihf Jfiih/k jiTc,;: Ihol hivs 7?etu'^-n hi ri,h^ Ir.i) then. 
Ihi'iilr hlw joint inl/i III. ii,.\i i(ii,rr thiit i.^' hetiiVt'ii /? ,6 
nn«' III II . or (i ._v, A','. 



/"A/A //-. 

/'■/>/. A. 





'^'-^ RAKING MOULDINGS, &c. 113 

Hi these black lines will exactly coincide with the front of the rib when bent round ; a line 
being drawn by the other edge of the moulds, will be perpendicular over its "plan, and the 
thickness of the sash-frame towards the inside will be found near enough by gauging from the 
outside. 



PLATE LXXVI. 

Qonstruction of a circular headed architrave is a circular wall. See the description on the 
plate. 

PLATE LXXVIL 

^" • To describe the Angle Bars for Shop Fronts. 

In Jig. A, B is a common bar, and C is the angle bar of the same thickness; take the 
raking projection 1, 1, in C, and set the foot of your compass in 1 at B, and cross the middle 
of the base at the other 1 ; then draw the lines 2, 2 ; 3, 3, &c. parallel to 1, 1 j then prick 
your bar at C from the ordinates so drawn at B, which being traced will give the angle bar. 

How to draw the Mitre Angle of a Commode Front for a Shop. 

In E divide the projection each way in a like number of equal parts, then the parallel lines 
continued each way will give the mitre. 

How to find the Raking Mouldings of a Pediment. 

^nfig. F\et the simarecta on the under side be the given moulding, and let lines be drawn 
upon the rake at discretion ; but if you please, let them be equally divided upon the simarecta, 
and drawn parallel to the rake ; then the mould at the middle being pricked oif from these level 
lines at the bottom, will give the form of the face. The return moulding at the top must be 
pricked upon the rake, according to the letters. 

[15] 






114 * SIMILAR MOULDINGS, &c. 

The cavettOj, fig. .C, is drawn in tlie same manner. 

N. B. If the middle moulding, fig. F, be given, perpendiculars must be drawn to the top 
of it; then horizontal lines must be drawn over the mouldings at each end,, with the same di- 
visions as are over the mouldings ; and lines being drawn perpendicularly down, as above, will 
show how to trace the end mouldings. 



PLATE LXXVIII. 

Raking Mouldings, and Mouldings of different Projections. 

Figures B and A show how to trace base mouldings for skirting to stairs, upon the same 
principles as shown in the last plate; at the bottom are given two methods of mitring mould- 
ings of different projections together. 



PLATE LXXIX. 

Given the Form of a Cornice, to draw it to a greater Proportion. 

In fig. Jl, let the given height of the cornice be a c? ; set one foot of your compass in «, 
and cross the under side at h with that height, and from the point c draw the line c ^ at right 
angles to a 6 ; then the height of all your mouldings will be the parts of a b, and the projec- 
tions the parts of c c? in proportion. 

JVote, af shows another height, c e its projection in proportion to that height. 

How to diminish a Cornice in the Proportion of a greater. 

Describe equilateral triangles on the base and projection as at D, and make if and i g 
equal to the intended height, and draw the line /^ across the triangle, which will give the 



rial, 




V 



y 



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^ 



/ 



I 



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M''^' i 



^ . 



^-^-'V 



Tlate 30. 



Fiff.^. 



mi'^. 





Tfii.s- Tlf/tr .fJtow 



Fui. C 



\ fo sta?ii? Yo tinv S/rrr/ia. l 






w^ 



SIMILAR MOULDINQS, &c. 115 

heights in proportion to ab; put the foot of your compass in 6 as a centre, and circle b c 
round b h, and draw the dotted line h i, cutting/^ in k ; then set off i e and i d, each equal to 
g k ; draw d e ; then take the divisions of e d, and set them from/ to m ; in the same order 
draw perpendiculars : it will give the diminished cornice at I). 

Another Method. 

At E, let the given height be a b, and draw the hypothenuse a g, and lines being 
squared up to a b, from the divisions of a g, will give the heights ; and if you draw the 
line ^ c? at a right angle with a g, then d c will give the projection in proportion, when return- 
ing upon d c. . 

Fig. C is the Method for hanging a Jib Door. 

Let a c be the projection of the surbase or base moulding, and c the centre of the hinge ; 
make a b equal to a c, and in the centre at c describe the arch b de', then the arch b d e will 
be the proper joint for the surbase to work in. 

The joint of the surbase or the base may also be straight, as you see by the dotted line 
touching the circle at the point b, as the tangent to it. 



PLATE LXXX. 



MOULDINGS UPON THE SPRING. 

To find ike Sweep of a Moulding to be bent upon the Spring round a circular Cylinder. 

In fig. A, which stands upon a semicircular plan, make ac equal to the height of your 
moulding, and make a b equal to the projection ; describe the form of the moulding, and draw 
a dotted line to touch the face of it ; then draw the line edio meet in the centre of the body 
at d, so as to keep your moulding to a sufficient parallel thickness ; from the centre d de- 
scribe the several concentric circles which are the arrises of the moulding required. 

15* 



116 SKY-LIGHTS. 

How to find the Sweep of your Moulding when the Plan is a Segment. 
Complete the semicircle as in plate 1, fig. 1, then proceed as described mfig. A. 

Figures C and Z> show the method for bending a moulding round the inside, which is 
performed the same as above. 

The demonstration may easily be conceived from the covering of a cone. 



PLATE LXXXL 

To find the Length of the Hips of a Sky -light standing upon a square Plan, the Height being 

given. 

In fig. Jl, draw the diagonals a b, and cd; they will bisect each other at right angles at e ; 
take a e for the base of any hip ; from ein e d make ef equal to the height of the sky-light ; 
join a, f, and af will be the length of the hip required. 

To find the Backing of the Hip. 

Draw any line k i, at right angles to a e, the base of the hip rafter, cutting it in any point 
hf put the foot of your compass in A, as a centre, and with the other describe a circle to 
touch af, the hip rafter, to cut the base line a e, at ^ ; then draw g i and g k ; then the angle 
k g i will be the backing of the hip, as is shown by the bevel at B ; but the best way to work 
the hips is to apply a bevel to the parallel sides of the hips, as is shown at C, by making the 
other side of the bevel parallel to a e, the base of the hip. 

JVote. The same lines will extend to any sky-light, whatever maybe the form of its 
plan ; if it be any polygon, to find the length of the hip rafter, draw a line through any point 
in its base at right angles to it, so as to cut the two contiguous sides to that base, and on the 
sliid point as a centre describe a circle to touch the hip rafter from the point where this cir- 
cle cuts the base line, draw two lines to meet the ends of the perpendicular line at the sides 
of the polygon ; then the angle formed by these two lines will be the backing required : but 
perhaps a few more examples will make it plainer than many words can. 



F/ny, //J. 



lP 



/}'/. A. 



Fnj. B. 







/>,/. D. 







w 



SKY-LIGHTS. 117 

Fig. jB is a sky-light, standing upon a rectangular base, having a ridge in the middle ; 
make c d upon the ridge line equal to half the width of a 6 ; then the angle bda will be a right 
angle : every other requisite is the same as directed for fig. A. If these hips are to be mitred, 
the bevel at C shows the mitre. 

Fig. C is another sky-light, standing also upon a rectangular base ; but the hips all meet 
over the centre of the plan at e, and consequently the diagonals do not bisect each other at 
right angles ; therefore take any base line as ae, or c g, and make ef perpendicular to a e, 
from e, and equal to the height of the sky-light , and draw /a or fg, for the length of the hip, 
by drawing the line I in at right angles to a e. The backing will be found in the same manner 
as the others above. This sky-light will require two different bevels D and £!, to be applied 
to the parallel sides of the hip, which are both found from the backing by drawing the stocks 
of the bevel parallel to a e, the base of the hip. 

But if the hips are to be mitred together, F and G show the two bevels for the mitring 

each half, so that when put together shall form the proper backing. 
$ 

Fig. Z) is a sky-light standing upon an octangular plan, as is described in fig. 8, plate 6, of 
the Geometry ; the lengths of the hips and backing of the angle are found in the same manner 
as directed for others. 

Fig. jEJ is a sky-light whose plan is trapezoid ; upon each end as a diameter describe a 
semicircle to cut the ridge line ; from these points draw lines to the extremities of their re- 
spective diameters, which will form a right angle for the hasp of the hips to stand upon ; the 
backing or mitring of the hips will be found as is described in fig. A and B. 



y^^. 




s^"' 



CONCLUSION 



In which are examined, by xvay of preventive Caution to the Student, 
several Methods, which are founded on wrong Principles, and 
better ones are here proposed. 



PLATE LXXXII. 

Of the Ellipsis. 

THE old manner for intersecting all kinds of lines, applied to Gothic and elliptical 
figures, to this day is exceedingly useful in formihg the ramps of stairs, or easing of any an- 
gle, as at G ; but when this is applied to elliptical figures, it is far from forming a true ellipsis, 
being too full at the ends, as at j^^. D and E ; and this is no certain rule for drawing an ellip- 
sis, for the more divisions there are, the worse is the ellipsis ; as for example, fig. F is divid- 
ed into double the number of parts as -D ; it is plain that neither D nor F is an agreeable 
ellipsis, and F is much worse than D, which is contrary to general opinion ; for I have been 
frequently told, the more parts it is divided into, the truer it is : but by this it appears the more 
parts it is divided into, the worse it is : if this is doubted, try. Fig. A is an ellipsis drawn on 
true principles, as laid down in plate 7, at C, of this book, and is here repeated to be com- 
pared with the others. Figures B, C, and E, are ellipses drawn with a compass : I may call 
them rejiresentations, as it is impossible to draw them true with a compass; there is no part 
of the curvature of an ellipsis that will exactly agree with any part of a circle, for in every 
quarter of an ellipsis, from the extremiity of the transverse, the curvature in every succeeding 



r"-^. 



120 CONCLUSION. 

part is continually flatter towards the extremity of the conjugate axis ; but yet there is a me- 
thod to represent an ellipsis, which will differ very little from the truth, as is shown at figures 
B and E, which are both drawn by the same method ; fig. JE or JB is the nearest to the shape 
of ^^. j1 ; fig. C, the method used by almost every author who has written upon the subject, 
as full at the ends, but not in so great a degree as fig, D and E. 



PLATE LXXXIII. 

Of Raking Mouldings. 

In plate 77, fig. t>5, let the moulding at the bottom be given, and let the perpendicular 
height be divided into any number of equal parts at 6 ; likewise divide the perpendicular 
height of the top moulding into six ; and the face moulding into six, at right angles to the 
rake j and let the ordinates of each be drawn through the equal divisions of each respective 
perpendicular, and pricked from the bottom, as the figures direct. It is evident that if the 
under moulding is composed of two quarters of a circle, the upper mouldings will be composed 
of two quarters of an ellipsis ; consequently the return moulding at the top will be too quick 
upon the round, and likewise in the hollow. 

But if this demonstration should not be sufficient, let us suppose a plane parallel to the 
arrises of the moulding, and perpendicular to the plane of projection, to pass through the 
point 1, this plane will be represented by a straight line 5 therefore let a dotted line be con- 
tinued from 1, in the given moulding, parallel to the rake, then it will be evidently seen, that 
this dotted line corresponds with neither the face nor the return moulding ; for in the face 
moulding it falls between the points 1 and 2 ; and in the top moulding it falls almost at the 
point 2 ; whereas it should only come to the point 1 in each ; but the horizontal projection 
from 2 to 2, at the top, ought to be equal to 1 1 at the bottom ; but it is much greater ; there- 
fore this method is false, and they will not mitre together. 

I shall also notice another method used by some authors, see figure B, at E and Ef where 
they are pricked perpendicular to their chords, in the middle, which is also false ; but if they 
are pricked as at B and C, on the rake, will be exceedingly near, if described with a compass 
through three points. 



^i:* 



f/nff //.> 



Fr<j. A. 



Ii.j.B. 




F/i/. C 



FLitfS^. 












^1 



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^ 

<1^ 



V 

— ^«- 



V 



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a 



a 



CONCLUSION. 121 

PLATE LXXXIV. 

Of diminishing of Columns. 

The method for drawdng a column, described by some authors, and which is properly 
called a conchoid column, is not only very inconvenient on account of the cumbersome instru- 
ment which is necessary to find the curve for practice, but also the appearance is, I think, less 
graceful than when produced by other methods. The conchoid curve, or column, is concave 
towards the bottom, and convex towards the top ; and if this curve was infinitely extended, it 
would never meet the axis ; which shows it to be different from the elliptic curve or column, 
as some have called it. 

The column called hyperbolic, plate 18, is not so named from the general properties of 
the conic hyperbola (because there may be an infinite number of hyperbolas standing upon the 
same base, having one common vertex, which will all be contained between a triangle and a 
parabola, according as its axis is longer or shorter,) but because it will nearly coincide with 
some of these hyperbolas. This curve has been known among workmen, and by them has been 
mistaken for an elliptic curve ; to refute which I have, on the same plate, shown a true elliptic 
column for comparison ; the lines of their curvature are continued only to show their true 
figure ; either of these is a more commodious method than the conchoid. 

The method which I recommend as easiest in practice for diminishing of columns is already 
described on plate 69, by means of a diminishing rule, which is infinitely more convenient than 
the trammel, and which to my eye, also produces a pleasanter contour : but as this will depend 
on the fancy of the architect, the workman will find some of the methods shown will answer his 
purpose for any curve. The conchoid is flattest at the top, the hyperbolic is a little quicker, 
the parabolic is still more so, and the elliptic is the most quick. 

[16] 



--■J? 



INDEX. 



PAGE 



ACUTE angle, defined 
Acute angled triangle, defined 
Altitude of a figure, defined - - - 
Angle, acute, defined . - - - 
Angle bracket . . - - - 
Angle, definition .... 

Angle, how bisected - - - - . 
Angle, how denoted . - . - 
Angle, how made equal to a given angle - 
Angle, how measured - - - - 
Angle, obtuse, defined ... 

Angle bars for shop fi-onts ... 

Angle ribs 

Angle, right, defined - - - ' - 
Angle mould of a groin - - - 

Angles, kinds of - - - - - 
Arc of a circle, defined ... 

Arc of a circle, how found by points 
Arc of a circle, how its length may be found 
Arch bar - - - - - ' - 
Arches, how described ... 

Ascending groins .... 



B 



Bevelling the edges of ribs 



Boarding of polygonal figures 



PAGE 

57 



10 


Boarding of domes, when the boards 


are 


bent 




11 


vertically - - - - 


- 


- . 


57 


12 


Boarding of domes, when the boards 


are 


bent 




10 


horizontally . - . 


- 




68 


51 


Boarding of an ellipsoidal dome 


- 


- 


60 


10 


Body, or body range of a groin 


- 


- 


36 


13 


Butt joints in hand-railing, how made 


- 


94, 


101 


10 










13 


C 








12 


Capping an iron rail ... 


- 


- 


100 


10 


Centre of a circle, defined 


- 


- 


11 


113 


Centring to groins - - - 


- 


36, 37 


45 


Chord in a circle, defined 


- 


- 


12 


10 


Circle, defined .... 


- 


- 


11 


36 


Circular work .... 


- 


- 


111 


10 


Circumference, defined 


- 


- 


11 


11 


Circumference of a circle, how described through 




20 


three given points 




- 


14 


15 


Columns, how diminished 




- 


121 


112 


Common bracket ... 




- 


51 


34 


Commode front - - - . 




- 


113 


37 


Cone, defined - - - . 




- 


19 




Conic sections .... 




- 


19 




Cornices, proportioned 




- 


114 


35 


Cot, or cod-bar .... 




. 


112 



16* 



124 



INDEX. 



• • PAGE 

Cove bracket - ■ ■ . ■ " " ^^ 

Covering spherical domes ... 57, 58 

Coves - - - - - - - -51, 

Cradling lo groins .... 37,46 

Cradling for the heads of niches - - 47, 50 

Cradhng pendentives .... 61, 52 

Curb, how formed in the ceilings of churches - 34 
Curtail step, how drawn .... 87 

Curve line, deSned - ... - 10 

Cylindric sections - - - - 22, 23 

Cylindric groins - - - - - - 39 

Cylindro-cylindric arch, defined - - - 43 

Decagon, defined - - - - - 11 

Designs for roofs - - - - 54 10 59 

Diameter of a circle, defined - - - 12 

Diminishing rule - - - - - 110 

Diminution of columns - - - - 110 

Dodecagon, defined - - - - - 11 

Dog-leg stairs -..-.. 86 

E 

Ellipsis, how circumscribed about a rectangle 
with its axis parallel to, and in the same 
ratio, as the sides of the rectangle - 17 

Ellipsis, a conic section - - - - 19 

Ellipsis, how described from the cone - - 19 
Ellipsis, how described by the intersection of 

straight lines _ .... 20 
Ellipsis, the two axes of an, being given, to de- 
scribe the curve with compasses, by ordi- 
nates, by a string, and by a trammel - 17 
Ellipsis, being given, to find its axis and centre 17 



I,- PAGE 

Ellipsis, the^ length of an, being given, to find the 
other axis, so that the two shall be in a gi- 
ven ratio .--.-. 18 
Ellipsis, observations thereon - - « HQ 
Ellipsdidal-headed niche - - - - 50 
Elliptic planed stair-case, how to draw the face 

and falling moulds - - - 97 
Endecagon, defined - - - - - 11 
Equilateral triangle, defined - - - 11 
Equilateral triangle, a regular figure - - 11 
Equilateral rectangle, defined - - - 11 
Equilateral parallelogram, defined - - 11 
Equilateral triangle, how constructed upon a gi- 
ven line - •• - . - - 13 



Face mould for the twist part of a scroll, how 

drawn - - - - - - 87 

Face mould for a semicircular turning round a 

level landing ..... 102 

Falling mould, how drawn - . - - - 87 

Four sided figures, how called - - - 11 



G 



Geometry 
Globe, its sections 
Groins, how formed 



H 



Hand-railing - 

Height of a figure, defined 

Hip roof, how constructed 

Hip sky-lights .... 

Hyperbola constructed from the cone 



9 to 27 

24 

36 to 43 



86 to 107 

12 

54 

- 115 

20 




INDEX. 



125 



Intersection of the angles of a groin 
Iron rail, how capped 
Irregular polygon, how constructed 
Isosceles triangle _ - - 



Jack rafter, how fitted upon the hip 
Jack ribs of a groin 
Jib door, how hung 



Kirb lights for" church work 



Level landing . . - - - 

Line, bisected _ . - - - 

Line, defined - - - - - 

Line divided as another - - . 

Line of nosings . . - . 

Lines, the several kinds ... 

Lining of a soflSt - - - - 

M 

Mean proportion, how found between two 

straight lines - - . . 

Mitre bracket of a cornice - -. - 
Mitre cap of a hand-rail, how formed 
Mouldings upon the spring ... 



PAGE 

39 

106 

15 

15 



55 

36, 37, 38 

- 115 



34, 35 



93, 102 
13 
10 

18 

- 104 

10 

29 



N 



Niches 



14 

51 

86,87 

115 



47,50 



Oblique lines - - - - 

Obtuse angle, defined . - - 
Obtuse angled triangle, defined 
Octagon, defined . . - 

Octagon made out of a given square 
Orthographical representation of a rail 



PAGE 

10 
10 
11 
11 

18 
91,92 



Parabola, a conic section 




- • 


19 


Parabola, how described from a cone 






19 


Parallel thickness of stuff, how found 




87,'^8 


Parallelogram, defined 






11 


Parallels, defined - . - 






10 


Pediment .... 






113 


Pendentives . . - - 






51 


Pentagon, defined ... 






11 


Pentagon, how made 






14 


Perpendicular, defined 






10 


Perpendicular, to draw a - - 






12 


Perpendicular, to let fall a - - 






13 


Perpendicular at the end of a right 


line, 






to draw a - - - - 






13 


Plane, defined .... 






11 


Plane figures, how bounded - 


> 




H 


Plane figures, how named 


- 




11 


Plane of the segment of a cylinder, defined 




22 


Point, definition of a - 


- 




10 


Polygon, defined . . - 


- 




11 


Polygon, irregular, defined 


- 




11 


Polygon, how constructed upon a given i 


right 




line - - . - . 


-, 


- 


14 


Polygon, irregular, being given to construct, 




and then equal and similar 


- 


- 


15 



126 



INDEX. 



Polygon roofs 

Purline, how fitted to the hip rafter- 

Q 

Quadrangle, defined - - - 
Quadrangle, how made equal to a given one 
Quadrant of a circle, defined 
Quadrilateral, defined - . - - 
Quoins of a groin . - . - 



R 



Raking mouldings 
Radial bars 



Radius of a circle, defined - - - - 

Ranging the edges of ribs standing upon curbs 
in churches - . . . . 

Ranging the ribs of groins 

Rectangle, defined . - - . . 

Rectangle, equilateral, defined ... 

Rectangle, how made equal to a given tri- 
angle - - - 

Rhomboid, defined ..... 

Rhombus, defined - - - . 

Rib of the front of the head of a niche 

Ribs of the back of the head of a niche - 

Ribs of thai spheric headed niches, how des- 
cribed ...... 

Ribs of spheric headed niches, how placed upon 
the front rib . , 

Ribs of spheric headed niches, how bevelled up- 
on the front rib - - - - - 

Right anglej defineil' ..... 

Right angled triangle, defined • - . - 

Regular pentagon, defined » - -. ' - 

Rule for diminishing columns ^ - *,- \ - 



PAGE 

S2, 33. 

54 



11 
15 
12 
11 

39 



120 

111 

12 



35 

40, 41, 42 
10 
■11 

15 
11 
11 

47 
48 



•••48 

49 

50 

10 

11 

•■ll 

110 



Roof in lodgement 

Roofing 

Roofs, designs for 



S 



Sash head, circular in circular wall 

Scalene triangle, defined - . _ 

Scroll of a hand-rail, how drawn 

Seat of the quoins of a groin 

Section of a hemisphere cut by a cylinder 
surface at right angles to the plane 
of the segment ■ - - . . 

Section of a cylinder, defined 

Section of a sphere, what .... 

Section of any body generated by rotation 
round a fixed axis - . - 

Section of a hemisphere cut by a plane at 
right angles to the plane of the segment 

Sections of a cyhnder .... 

Section of a hand-rail, how formed 

Sections of a sphere - - - 

Sector of a circle, defined ... 

Segment of a circle, defined 

Segment of a circle, how constructed to 
any chord, and versed sine by a com- 
pass, or by the intersection of straight 
lines ...--- 

Segment of a cylinder being given, to cut ^ 
it by a plane through a given line on 
the plane of a segment, and to make 
any given angle with' that plane 

Semicircle, defined .... 

Skirting, how fitted down - ,* - 

^Sky-lights - - 

Soffits, linings of - ^ - - - 



PAGE 


55 


54 to 61 


79 to 83 


- Ill 


11 


86 


36 



24,25 

21,22 

24 

24 

25 
21 
85 
24 
12 
12 



16,20 



• ' 21 
12 
- 109 
116, 117 
29 to 33 






0' 



INDEX. 



Solid, definition of a - 

Spherical domes _ - . . 

Spheric sections . . . ■ 

Spheric niches - - - - - 
Springing of a pendentive - - - 
Spheroidal domes - . - . 

Spring of the plank . - . - 
Square, defined • • - 
Square, how made upon ^ given right line 
Square, how made equal to a given rect- 
angle . ■ - - . 
Square, how made equal to two given 

squares - - - " - 
Square, how made equal to three given 

squares - - . 
Square, being given, to cut off its angles 
so as to form an octagon - . - 
Stair-casing . . . . . 

Straight line, defined - - - - 
Stretch-out line - - - . - 
Strength of timber - - . . 

Superfices, kinds of - - . - 
S.uperfices, definition of - - . 



PAGE 


10 


36 to 60 


24 


' 47 to 49 


61 


60,61 


- 100 


11 


13 


13, 15, 16 


• 


16 


16 


11 


85 to 107 


10 


29 


63 to 78 


10 


10 



Superfices, pla.io. definition 
Superfices, cuj ved - ■' 

T . 

Tangent, defined - , - - • 1) 

Cangent to a circle, how drawn * >» - - 14 

... ' 
Tangent to a circle bemg given, to find 

the point of contact - - - - 14 

Theoretical part of Geometry, how founded - 9 

Thickness of stuff 106 

Trapezium, defined - - - - - 11 

Trapezoid, defined - - - - - 11 

Triangle, equilateral, defined - - - 11 

Triangle, isosceles, defined - - - - 11 

Triangle, scalene, defined - - - - 11 

Triangle, right angled, definition of a - - 11 

Triangle, oblique, definition of a - - - 11 

Triangle, acute angled, definition of a - - 11 

Triangle, obtuse angled, definition of a - - 11 
Triangle, how constructed of three given 

right lines --.-.. 15 

Triangles, kinds of, definition - - 34 to 43 

Truss girders for floors .... 80 






/ 



L ■ 



, V 



